Lattices and Ordered Sets Steven Roman Lattices and Ordered Sets Dr. Steven Roman Irvine, CA 92603 USA ISBN: 978-0-387-78900-2 e-ISBN: 978-0-387-78901-9 DOI: 10.1007/978-0-387-78901-9 Library of Congress Control Number: 2008928921 Mathematics Subject Classification (2000): 06-xx, 03-xx, 03Exx, 04-xx © 2008 Steven Roman All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com To Donna and to my friend Gian-Carlo Rota, who is missed Preface This book is intended to be a thorough introduction to the subject of order and lattices, with an emphasis on the latter. It can be used for a course at the graduate or advanced undergraduate level or for independent study. Prerequisites are kept to a minimum, but an introductory course in abstract algebra is highly recommended, since many of the examples are drawn from this area. This is a book on pure mathematics: I do not discuss the applications of lattice theory to physics, computer science or other disciplines. Lattice theory began in the early 1890s, when Richard Dedekind wanted to know the answer to the following question: Given three subgroups E, F and G of an abelian group K, what is the largest number of distinct subgroups that can be formed using these subgroups and the operations of intersection and sum (join), as in E Fß ÐE FÑ ∩ Gß E ÐF ∩ GÑ and so on? In lattice-theoretic terms, this is the number of elements in the relatively free modular lattice on three generators. Dedekind [15] answered this question (the answer is #)) and wrote two papers on the subject of lattice theory, but then the subject lay relatively dormant until Garrett Birkhoff, Oystein Ore and others picked it up in the 1930s. Since then, many noted mathematicians have contributed to the subject, including Garrett Birkhoff, Richard Dedekind, Israel Gelfand, George Grätzer, Aleksandr Kurosh, Anatoly Malcev, Oystein Ore, Gian-Carlo Rota, Alfred Tarski and Johnny von Neumann. This book is divided into two parts. The first part (Chapters 1–9) outlines the basic notions of the theory and the second part (Chapters 10–12) contains independent topics. Specifically, Chapter 1 describes the basic theory of partially ordered sets, including duality, chain conditions and Dilworth's theorem. Chapter 2 is devoted to the basics of well-ordered sets, especially ordinal and cardinal numbers, a topic that is usually presented only in classes on set theory. However, I have included it here for two reasons. First, we will have a few occasions to use viii Preface ordinal numbers, most notably in describing conditions that characterize the incompleteness of a lattice and in the discussion of fixed points. Second, it is my experience that many students of algebra do not take a class in set theory and so this seems a reasonable place to get a first exposure to this important subject. In any case, the chapter may be omitted upon first reading without severe consequences: Only a few theorems will be inaccessible to readers with no background in the subject. However, I have also included nonordinal versions (for countable lattices) of these theorems. Chapter 3 begins the study of lattices, introducing the key properties such as join-irreducibility, completeness, denseness, lattice homomorphisms, the F down map, ideals and filters, prime and maximal ideals and the DedekindMacNeille completion. Chapter 4 is devoted to the basics of modular and distributive lattices, including the many characterizations of these important types of lattices and the effect of distributivity on join-irreducibility and prime ideals. Chapter 5 is devoted to Boolean algebras. This is a vast field, with many applications, but we concentrate on the lattice-theoretic concepts, such as characterizing Boolean algebras as Boolean rings, or in terms of weak forms of modularity or de Morgan's laws. We also discuss complete and infinite distributivity. Chapter 6 concerns the representation theory of distributive lattices. We describe the representation of a distributive lattice with the DCC as a power set sublattice, the representation of complete atomic Boolean algebras as power sets and the representation of an arbitrary distributive lattice as a sublattice of downsets of prime ideals. Chapter 7 is devoted to algebraic lattices, a topic that should be of special interest to all those interested in abstract algebra. Here we prove that a lattice is algebraic if and only if it is isomorphic to the subalgebra lattice of some algebra. Chapter 8 concerns the existence of maximal and prime ideals in lattices. Here we discuss various types of separation theorems and their relationship with distributivity, including the famous Boolean prime ideal theorem. This material is key to an understanding of the Stone and Priestley duality theory, described in a later chapter. Chapter 9 is devoted to congruence relations on lattices, quotient lattices, standard congruences and sectionally complemented lattices. Chapters 10–12 are topics chapters. They can be covered in any order. Chapter 10 is devoted to the duality theory for bounded distributive lattices and Boolean algebras. We cover the finite case first, where topological notions are not required. Then we discuss the necessary topics concerning Boolean and ordered Boolean spaces. Although very little point-set topology is used, a brief appendix Preface ix is included for readers who are not familiar with the subject. In both the finite and nonfinite cases, the dualities are most elegantly described using the language of category theory. For readers who are not familiar with this language, an appendix is provided with just enough category theory for our discussion in the text. (It is not so much the results of category theory that we need, but rather the language of category theory.) Chapter 11 is an introduction to free lattices and Chapter 12 covers fixed-point theorems for monotone and for inflationary functions on complete partially ordered sets and complete lattices. Contents Preface Contents Part I: Basic Theory 1 Partially Ordered Sets, 1 Basic Definitions, 1 Duality, 10 Monotone Maps, 13 Down-Sets and the Down Map, 13 Height and Graded Posets, 15 Chain Conditions, 16 Chain Conditions and Finiteness, 17 Dilworth's Theorem, 18 Symmetric and Transitive Closures, 20 Compatible Total Orders, 21 The Poset of Partial Orders, 23 Exercises, 23 2 Well-Ordered Sets, 27 Well-Ordered Sets, 27 Ordinal Numbers, 31 Transfinite Induction, 37 Cardinal Numbers, 37 Ordinal and Cardinal Arithmetic, 42 Complete Posets, 43 Cofinality, 46 Exercises, 47 3 Lattices, 49 Closure and Inheritance, 49 Semilattices, 51 Arbitrary Meets Equivalent to Arbitrary Joins, 52 Lattices, 53 Meet-Structures and Closure Operators, 55 xii Contents Properties of Lattices, 60 Join-Irreducible and Meet-Irreducible Elements, 64 Completeness, 65 Sublattices, 68 Denseness, 70 Lattice Homomorphisms, 71 The F -Down Map, 73 Ideals and Filters, 74 Prime and Maximal Ideals, 77 Lattice Representations, 79 Special Types of Lattices, 80 The Dedekind–MacNeille Completion, 85 Exercises, 89 4 Modular and Distributive Lattices, 95 Quadrilaterals, 95 The Definitions, 95 Examples, 96 Characterizations, 98 Modularity and Semimodularity, 104 Partition Lattices and Representations, 110 Distributive Lattices, 121 Irredundant Join-Irreducible Representations, 123 Exercises, 124 5 Boolean Algebras, 129 Boolean Lattices, 129 Boolean Algebras, 130 Boolean Rings, 131 Boolean Homomorphisms, 134 Characterizing Boolean Lattices, 136 Complete and Infinite Distributivity, 138 Exercises, 142 6 The Representation of Distributive Lattices, 145 The Representation of Distributive Lattices with DCC, 145 The Representation of Atomic Boolean Algebras, 146 The Representation of Arbitrary Distributive Lattices, 147 Summary, 149 Exercises, 150 7 Algebraic Lattices, 153 Motivation, 153 Algebraic Lattices, 155 Ä ∩ ∪ -Structures, 156 Algebraic Closure Operators, 158 The Main Correspondence, 159 Subalgebra Lattices, 160 Contents xiii Congruence Lattices, 162 Meet-Representations, 163 Exercises, 166 8 Prime and Maximal Ideals; Separation Theorems, 169 Separation Theorems, 169 Exercises, 176 9 Congruence Relations on Lattices, 179 Congruence Relations on Lattices, 180 The Lattice of Congruence Relations, 185 Commuting Congruences and Joins, 187 Quotient Lattices and Kernels, 189 Congruence Relations and Lattice Homomorphisms, 191 Standard Ideals and Standard Congruence Relations, 195 Exercises, 202 Part II: Topics 10 Duality for Distributive Lattices: The Priestley Topology, 209 The Duality Between Finite Distributive Lattices and Finite Posets, 215 Totally Order-Separated Spaces, 218 The Priestley Prime Ideal Space, 219 The Priestley Duality, 222 The Case of Boolean Algebras, 229 Applications, 230 Exercises, 235 11 Free Lattices, 239 Lattice Identities, 239 Free and Relatively Free Lattices, 240 Constructing a Relatively Free Lattice, 243 Characterizing Equational Classes of Lattices, 245 The Word Problem for Free Lattices, 247 Canonical Forms, 250 The Free Lattice on Three Generators Is Infinite, 255 Exercises, 259 12 Fixed-Point Theorems, 263 Fixed Point Terminology, 264 Fixed-Point Theorems: Complete Lattices, 265 Fixed-Point Theorems: Complete Posets, 269 Exercises, 274 A1 A Bit of Topology, 277 Topological Spaces, 277 Subspaces, 277 Bases and Subbases, 277 xiv Contents Connectedness and Separation, 278 Compactness, 278 Continuity, 280 The Product Topology, 280 A2 A Bit of Category Theory, 283 Categories, 283 Functors, 285 Natural Transformations, 287 References, 293 Index of Symbols, 297 Index, 299 Part I: Basic Theory Chapter 1 Partially Ordered Sets Basic Definitions We begin with some basic definitions. System of Distinct Representatives We will have occasions to use the following basic concept. Definition Let Y œ ÖE3 ± 3 − M× be a family of sets. A system of distinct representatives or SDR for Y is a set Ö+3 ± 3 − M× of distinct elements with the property that +3 − E3 for all 3. Cartesian Products The cartesian product of two sets W and X is generally defined as the set W ‚ X œ ÖÐ=ß >Ñ ± = − Wß > − X × of ordered pairs. However, this does not easily generalize to the cartesian product of an arbitrary family of sets. For this, we use functions. Definition The cartesian product of a family Y œ Ö\3 ± 3 − M× of sets is the set $Y œ $\3 œ š0 À M Ä .\3 ¹ 0 Ð3Ñ − E3 › 3−M Binary Relations Definition Let E be a nonempty set. A binary relation on E is a subset V of the cartesian product E ‚ E. We write Ð+ß ,Ñ − V as + µ , . A binary relation is S. Roman (ed.), Lattices and Ordered Sets, doi: 10.1007/978-0-387-78901-9_1, © Steven Roman 2008 1 2 Lattices and Ordered Sets 1) reflexive if for all + − E, +µ+ 2) irreflexive if for all + − E, +µ y + 3) symmetric if for all +ß , − E, +µ, Ê ,µ+ Ê ,µ y + 4) asymmetric if for all +ß , − E, +µ, 5) antisymmetric if for all +ß , − E, + µ ,ß ,µ+ Ê +œ, ,µ- Ê +µ- 6) transitive if for all +ß , − E, + µ ,ß By far the two most important combinations of these axioms are found in the definition of an equivalence relation and a partial order. Definition An equivalence relation on a nonempty set E is a binary relation on E that is reflexive, symmetric and transitive. If + − E, the set Ò+Ó of all elements of E that are equivalent to + is called the equivalence class containing +. If X is an equivalence relation on a set \ , then a system of distinct representatives for the equivalence classes of X is also called a system of distinct representatives for X . Partial Orders and Posets We can now define one of the principal objects of study of this book. Definition A partial order (or just an order) on a nonempty set T is a binary relation Ÿ on T that is reflexive, antisymmetric and transitive, specifically, for all Bß Cß D − T : 1) (reflexive) BŸB 2) (antisymmetric) B Ÿ Cß CŸB Ê BœC B Ÿ Cß CŸD Ê BŸD 3) (transitive) The pair ÐT ß Ÿ Ñ is called a partially ordered set or poset, although it is often said that T is a poset, when the order relation is understood. If B Ÿ C, then B is Partially Ordered Sets 3 less than or equal to C or C is greater than or equal to B. We also say that B is contained in C or that C contains B. If B Ÿ C but B Á C, we write B C or C B. If B Ÿ C or C Ÿ B, then B and C are said to be comparable. Otherwise, B and C are incomparable or parallel, denoted by + ² , . If W and X are subsets of a poset T , then W Ÿ X means that = Ÿ > for all = − Wß > − X . If X œ Ö>×, then W Ÿ Ö>× is written W Ÿ > and similarly for = Ÿ X . Definition A poset ÐT ß Ÿ Ñ is totally ordered if every Bß C − T are comparable, that is, BŸC or C Ÿ B In this case, the order is said to be total or linear. We will assume unless otherwise noted that a nonempty subset W of a poset T inherits the order from T , making W a poset as well. We will have some use for the following concept in the exercises. Definition A preorder or quasiorder on a nonempty set T is a binary relation that is reflexive and transitive. Example 1.1 1) Let œ Ö!ß "ß á × be the set of natural numbers. Then Ðß Ÿ Ñ is a poset under the ordinary order. Also, Ðß ± Ñ is a poset under division, that is, where B ± C means that C œ 5B for some 5 − , that is, B divides C . 2) If \ is a nonempty set, then the power set kÐ\Ñ of \ is the set of all subsets of \ . It is well known that kÐ\Ñ is a poset under set inclusion. 3) The set fÐKÑ of all subgroups of a group K is a poset under set inclusion. The set a ÐKÑ of all normal subgroups of K is a subset of f ÐKÑ, inheriting the order relation from fÐKÑ. Many other such examples are possible, involving other algebraic structures. 4) The set c of all partitions of a nonempty set \ is a poset, where - Ÿ 5 if is a refinement of 5, that is, if every block of 5 is a union of blocks of -. Put another way, the blocks of - are constructed by further partitioning some (or none) of the blocks of 5. 5) Let T be the collection of partial functions on a nonempty set \ , that is, functions on \ whose domain is a subset of \ . Order T by saying that 0 Ÿ 1 if 0 is an extension of 1, that is, 0 and 1 agree on the domÐ0 Ñ. The T is a poset under this order. 6) The product order on the cartesian product T ‚ U is defined by Ð:" ß ;" Ñ Ÿ Ð:# ß ;# Ñ if :" Ÿ :# and ;" Ÿ ;# The set T ‚ U with this order is called the product of T and U. 4 Lattices and Ordered Sets 7) The lexicographic order on the cartesian product T ‚ U is defined by Ð:" ß ;" Ñ Ÿ Ð:# ß ;# Ñ if :" :# or (:" œ :# and ;" Ÿ ;# ) This is also a partial order on T ‚ U. Strict Orders To every partial order on a set T there corresponds a strict partial order. Definition A binary relation on a nonempty set T is called a strict partial order (or strict order) if it is asymmetric and transitive. Given a partially ordered set ÐT ß Ÿ Ñ, we may define a strict order by BC if B Ÿ C and B Á C Conversely, if is a strict order on a nonempty set T , then the binary relation defined by BŸC if B C or B œ C is a partial order and so ÐT ß Ÿ Ñ is a poset. Thus, there is a one-to-one correspondence between partial orders and strict partial orders on a nonempty set T and so a partially ordered set can be defined as a nonempty set with a strict order relation. The Covering Relation We next define the covering relation on a poset. Definition Let ÐT ß Ÿ Ñ be a poset. Then C covers B in T , denoted by B ö C, if B C and no element in T lies strictly between B and C, that is, BŸDŸC Ê DœB or D œ C If + ö , or + œ , , we write + « , . For a finite poset T , the covering relation uniquely determines the order on T , since + Ÿ , if and only if there is a finite sequence of elements of T such that + ö :" ö : # ö â ö : 8 ö , Small finite posets are often described with a diagram called a Hasse diagram, which is a graph whose nodes are labeled with the elements of the poset and whose edges indicate the covering relation. This is illustrated in the following example. Example 1.2 Figure 1.1 shows the Hasse diagram of the poset Ö"ß #ß $ß 'ß "#× under division. Partially Ordered Sets 5 12 6 2 3 1 Figure 1.1 Figure 1.2 shows the Hasse diagrams of two posets x y x z y z N5 M5 Figure 1.2 that will play key roles in later discussions. Chains and Directed Sets Totally ordered subsets of a poset play an important role in the theory of partial orders. Definition Let ÐT ß Ÿ Ñ be a poset. 1) A nonempty subset W of T is a chain in T if W is totally ordered by Ÿ . A finite chain with 8 elements can be written in the form -" -# â -8 Such a chain said to have length 8 ". If the set of lengths of all chains contained in T is bounded, then T has finite length and the maximum length is called the length of T . Otherwise, T has infinite length. If + , , then a chain from + to , in T is a chain in T whose smallest element is + and whose largest element is , . A maximal chain from + to , is a chain from + to , that is not contained in a larger (in the sense of set inclusion) chain from + to , . 2) A nonempty subset W of T is an antichain in T if every two elements of W are incomparable. An antichain with 8 elements is said to have width 8. If the set of widths of all antichains in T is bounded, then T has finite width and the maximum width is called the width of T . Otherwise, T has infinite 6 Lattices and Ordered Sets width. A maximal antichain is an antichain that is not contained in a larger (in the sense of set inclusion) antichain. It is possible to combine chains when the largest element of one chain is the same as the smallest element of the other chain. Theorem 1.3 Let T be a poset. 1) If V+ß, is a chain from + to , and W,ß- is a chain from , to - , then V+ß, ∪ W,ßis a chain from + to - . 2) If V+ß, is a maximal chain from + to , and W,ß- is a maximal chain from , to - , then V+ß, ∪ W,ß- is a maximal chain from + to - . In some circumstances, the property of being a chain is stronger than is required and the following property is sufficient. Definition A nonempty subset H of a poset T is directed if every pair Ö+ß ,× of elements of H has an upper bound in H, that is, if for every +ß , − H, there is a - − H with the property that + Ÿ - and , Ÿ - . It is easy to see that a directed set T has the property that every finite subset of T has an upper bound in T . It is also clear that any chain is a directed set. Here is an example where a directed set is a more suitable choice than a chain: The union of every directed family of subspaces of a vector space Z is a subspace of Z . Similar statements can be made about subgroups of a group, ideals in a ring and so on. Maximal and Minimal Elements Maximal and maximum elements can be defined in posets. Definition Let ÐT ß Ÿ Ñ be a partially ordered set. 1) A maximal element is an element 7 − T that is not contained in any other element of T , that is, : − Tß 7Ÿ: Ê 7œ: A maximum (largest or greatest) element 7 in T is an element that contains every element of T , that is, :−T Ê :Ÿ7 We will generally denote the largest element by " and call it the unit element. 2) A minimal element is an element 8 − T that does not contain any other element of T , that is, : − Tß :Ÿ8 Ê :œ8 A minimum (smallest or least) element 8 in T is an element contained in Partially Ordered Sets 7 all other elements of T , that is, :−T Ê 8Ÿ: We will generally denote the smallest element by ! and call it the zero element. A partially ordered set is bounded if it has both a ! and a ". If a poset T does not have a smallest element, then we can adjoin one as follows. Let ! be a symbol that does not represent any element of T and let T¼ œ T “ Ö!× be the disjoint union of T and Ö!×. Extend the order in T to T¼ by including the condition that ! B, for all B − T . Similarly, a " can be adjoined by taking Tð œ T “ Ö"× and specifying that B " for all B − T . Definition If a poset T has a smallest element !, then any cover of ! is called an atom or point of T . The set of all atoms of a poset T is denoted by TÐT Ñ. A poset with ! is atomic if every nonzero element contains an atom. If T has a ", then any element covered by " is called a coatom or copoint of T . Upper and Lower Bounds Upper and lower bounds can be defined in a poset. Definition Let ÐT ß Ÿ Ñ be a partially ordered set and let W © T . 1) An upper bound for W is an element B − T for which WŸB The set of all upper bounds for W is denoted by W ? . We abbreviate Ö=×? by =? . If W ? has a least element, it is called the join or least upper bound or supremum of W and is denoted by 1W . The join of a finite set W œ Ö+" ß á ß +8 × is denoted by +" ” â ” +8 2) A lower bound for W is an element B − T for which BŸW The set of all lower bounds for W is denoted by W j . We abbreviate Ö=×j by =j. If W j has a greatest element, it is called the meet or greatest lower bound or infimum of W and is denoted by 3W . The meet of a finite set W œ Ö+" ß á ß +8 × is denoted by +" • â • +8 8 Lattices and Ordered Sets Note that the join of the empty set g is, by definition, the least upper bound of the elements of g. Since every element of T is an upper bound for g, we have 1g œ ! if T has a smallest element !; otherwise g has no join. Similarly, 3g œ " if " exists; otherwise 3g does not exist. Zorn's Lemma and Friends We will assume various equivalent forms of the axiom of choice. The Axiom of Choice Theorem 1.4 The following are equivalent: 1) (Axiom of choice) Let Y œ ÖW3 ± 3 − M× be a nonempty family of nonempty sets. Then there is a function 0 À Y Ä .3−M W3 for which 0 ÐW3 Ñ − W3 for all 3 − M . Such a function is called a choice function, since it chooses a single element 0 ÐW3 Ñ from each set W3 . 2) Every partition c of a set W has a system of distinct representatives. 3) The cartesian product C\3 of a set Y œ Ö\3 ± 3 − M× of nonempty sets is nonempty. Proof. To see that 1) implies 2), if c œ ÖF3 ± 3 − M×, then c is a set of sets and a choice function 0 gives a system Ö0 ÐF3 Ñ× of distinct representatives. To see that 2) implies 3), the set W œ ÖÐ3ß BÑ ± 3 − Mß B − \3 × can be partitioned using the first coordinate, that is, into the blocks F3 œ ÖÐ3ß BÑ ± B − \3 × for each 3 − M . A system of distinct representatives 0 œ ÖÐ3ß B3 Ñ ± 3 − M× for this partition belongs to the cartesian product #\3 , which is therefore nonempty. To see that 3) implies 1), let W be a set of sets. We may assume that no set in W is empty. Index each set using itself, that is, let \s œ = for all = − W . Then the indexed family Ö\= ± = − Wß = Á g× has a nonempty cartesian product, that is, there is a function 0 À W Ä -\= such that 0 Ð=Ñ − \= œ =. This is a choice function for W . Zorn's Lemma One of the most commonly used equivalences of the axiom of choice is Zorn's lemma. Theorem 1.5 The following are equivalent. Partially Ordered Sets 9 1) (Zorn's lemma) If a poset T has the property that every chain in T has an upper bound, then T has a maximal element. 2) If a poset T has the property that every chain in T has an upper bound, then for every : − T , there is a maximal element ; − T such that ; :. Proof. It is clear that 2) implies 1), since any maximal element ; − T such that ; : is a maximal element of T . Conversely, if 1) holds and if T has the property that every chain in T has an upper bound then :? also has this property and so 1) implies that :? has a maximal element ; , which is also maximal in T and satisfies ; :. The Well-Ordering Principle The well-ordering principle is also a commonly used equivalence of the axiom of choice. Definition A poset Ð[ ß Ÿ Ñ is a well-ordered set if every nonempty subset of W has a least element. In this case, the order Ÿ is called a well-ordering of [ . A nonempty set \ for which a well-ordering exists is called a well-orderable set. Note that a well-ordering is a total ordering, since if + Á , , then the set Ö+ß ,× © T has a least element. However, the usual ordering on ™ is a total ordering that is not a well-ordering. Definition (The well-ordering principle) Every nonempty set T is wellorderable. Tukey's Lemma and the Hausdorff Maximality Principle We will have relatively few occasions to use the following equivalences of the axiom of choice, so the reader may skip this now and return to it later when needed. Definition Let \ be a nonempty set. A family Y of subsets of \ has finite character if it has the property that E − Y if and only if every finite subset of E is in Y . An important example of a family of finite character is the family of all linearly independent subsets of a vector space. Definition (Tukey's lemma) Every nonempty family Y of sets of finite character has a maximal element. Definition (Hausdorff maximality principle) Let T be a poset. Any chain in T is contained in a maximal chain. Equivalences Let us now state formally that the previous statements are equivalent. 10 Lattices and Ordered Sets Theorem 1.6 The following are equivalent: 1) (Zorn's lemma) If T is a partially ordered set in which every chain has an upper bound, then T has a maximal element. 2) (Hausdorff maximality principle) Let T be a poset. Any chain in T is contained in a maximal chain. 3) (Tukey's lemma) Every nonempty family Y of sets that has finite character has a maximal element. 4) (Axiom of choice) Every set of sets has a choice function. 5) (Well-ordering principle) Every nonempty set can be well-ordered. Miscellaneous The following concepts are often useful. Definition Let ÐT ß Ÿ Ñ be a poset. For +ß , − T the closed interval from + to , is the set Ò+ß ,Ó œ ÖB − T ± + Ÿ B Ÿ ,× The initial segment up to + is the set Ð∞ß +Ñ œ ÖB − T ± B +× and the closed initial segment up to + is the set Ð∞ß +Ó œ ÖB − T ± B Ÿ +× A nonempty subset W of a poset T is convex if whenever +ß , − W , then Ò+ß ,Ó © W. Duality We now turn to a discussion of duality. Definition Let ÐT ß Ÿ Ñ be a poset. The dual poset ÐT ` ß £ Ñ is the poset with the same underlying set but whose order relation is the opposite of Ÿ , that is, B £ C if and only if B C. Of course, ÐT ` Ñ` œ T . Statements and Their Duals A statement about posets or a property of posets is just a statement that may or may not be true in any given poset. If = is a statement about posets, we write T Ð=Ñ to denote the statement = as applied to the poset T . If = is true in T , we say that T satisfies, possesses or has property = and write T } = (also read T models =). We can also view a property of posets as a subfamily of the family of all posets, namely, the subfamily consisting of all Partially Ordered Sets 11 posets in which = is true. If C is a set of properties, we say that a poset T satisfies C and write T } C if T has every property in C. Two statements = and > are logically equivalent if they are true in exactly the same posets, that is, if T }= Í T }> for all posets T . Let = be a statement about a poset T . One often defines the dual statement to = to be the statement formed by replacing all references (direct or indirect) to Ÿ with . Thus, least upper bound is replaced by greatest lower bound, maximal by minimal and so on. This is a bit vague, so we make the following definition. Definition Let = be a statement about posets. A dual statement =` is a statement about posets that satisfies the following property: T }= Í T ` } =` for all posets T . If C œ Ö:3 ± 3 − M×, then C` denotes any set of statements Ö:3` ± 3 − M×, where :3` is a dual to :3 . Note that if > and ? are dual statements to =, then T }> Í T` } = Í T }? for all posets T and so > and ? hold in exactly the same posets, that is, > and ? are logically equivalent. In fact, if = and =` are dual, then > is dual to = if and only if > is logically equivalent to =` . As mentioned earlier, one way to find a dual statement to = is to translate T ` Ð=Ñ into a logically equivalent statement about T , where T denotes an arbitrary poset. For example, if T Ð=Ñ is the statement T Ð=Ñ: If the join of a nonempty subset of T exists, it is unique then T ` Ð=Ñ: If the join of a nonempty subset of T ` exists, it is unique Since a join in T is a meet in T ` and vice versa, an equivalent dual statement is T Ð=Ñ` : If the meet of a nonempty subset of T exists, it is unique Definition A statement = about posets is self-dual if it is logically equivalent to any (and therefore all) of its dual statements =` , that is, if T }= Í T } =` A family C œ Ö:3 ± 3 − M× of properties of posets is self-dual if 12 Lattices and Ordered Sets T }C T } C` Í for all posets T . If a statement = holds in all posets satisfying C, that is, if T }C Ê T }= for all T , we write C Ê =. Now we can state the duality principle for posets. Theorem 1.7 (The duality principle for posets) Let C be a set of properties of posets and let = be a property of posets. Then C Ê = implies C` Ê =` C Ê = implies C Ê =` and if C is self-dual, then that is, if = holds in all posets satisfying C, then so does =` . In particular, g Ê = implies g Ê =` that is, if = holds in all posets, then so does =` . Let us give a simple example of the use of the duality principle. Theorem 1.8 1) Every finite poset with ! and with at least two elements has an atom. 2) Dually, every finite poset with " and with at least two elements has a coatom. Proof. For part 1), let T be a finite poset with ! and let +" − T be nonzero. If +" is not an atom, there is an element +# for which ! +# +" . If +# is not an atom, we may repeat the process to get ! +$ +# +" . Since T is finite, this process must stop to reveal an atom. Part 2) follows by duality. To illustrate how duality is applied in this case, let C œ Ö,ß 0 ß >× consist of the properties ,À has !ß 0 À finite and AÀ at least two elements Since T has ! if and only if T ` has ", we have , ` œ >À has " Of course, 0 and A are self-dual. Hence, C` œ Ö>ß 0 ß A× represents the family of all finite posets with ". Also, if +À has an atom then Partially Ordered Sets 13 +` À has a coatom Thus, the principle of duality gives CÊ+ implies C` Ê +` which is part 2). Monotone Maps We now define order-preserving maps. Definition Let T and U be posets and let 0 À T Ä U. 1) 0 is order-preserving or monotone or isotone if BŸC Ê 0 ÐBÑ Ÿ 0 ÐCÑ Ê 0 ÐBÑ 0 ÐCÑ Í 0 ÐBÑ Ÿ 0 ÐCÑ and strictly monotone if BC 2) 0 is an order embedding if BŸC (Note that such a map must be injective.) If 0 À T Ä U is an order embedding, we say that T can be order embedded, or simply embedded in U and write 0 À T ä U or just T ä U. 3) An order embedding 0 is an order isomorphism if it is also surjective. If 0 À T Ä U is an order isomorphism, we write 0 À T ¸ U or just T ¸ U. Note that if 0 À T Ä U is a monotone bijection, then 0 " need not be monotone, that is, 0 need not be an order isomorphism. (Map two incomparable elements to two comparable elements.) Down-Sets and the Down Map The following types of sets will occur frequently. Definition A subset H of a poset T is a down-set or order ideal if B − Hß CŸB Ê C−H We denote the down-sets in T by bÐT Ñ. Dually, a subset Y of T is an up-set or order filter if B − Yß C B We denote the up-sets in T by h ÐT Ñ. Ê C−Y 14 Lattices and Ordered Sets It is easy to see that H is a down-set in T if and only if its complement T Ï H is an up-set in T . Note also that the down-set property is transitive, that is, H − bÐIÑß I − bÐJ Ñ Ê H − bÐJ Ñ Theorem 1.9 Let T be a poset. 1) The union or intersection of any family of down-sets is a down-set. 2) The union or intersection of any family of up-sets is an up-set. That is, bÐT Ñ and h ÐT Ñ are closed under arbitrary union and arbitrary intersection. Theorem 1.10 Let W be a subset of a poset T . 1) The down closure of W is the smallest down-set containing W . This set is Æ W œ ÖB − T ± B Ÿ = for some = − W× and is read “down W .” The down-set Æ : œ Æ Ö:× is called the principal ideal generated by :. 2) Dually, the up closure of W is the smallest up-set of T containing W . This set is Å W œ ÖB − T ± B = for some = − W× and is read “up W .” The up-set Å : œ Å Ö:× is called the principal filter generated by :. The following maps are among the most important in the theory of order. Theorem 1.11 Let T be a poset. 1) The map 9Æ À T Ä kÐT Ñ defined by 9Æ Ð+Ñ œ Æ + œ ÖB − T ± B Ÿ +× is called the down map. The down map 9Æ is an order embedding of T into kÐT Ñ. Thus, any poset T can be order embedded in a power set kÐT Ñ. 2) The map 9Å À T Ä kÐT Ñ defined by 9Å Ð+Ñ œ Å + œ ÖB − T ± B +× is called the up map. The up map 9Å is an order anti-embedding of T into kÐT Ñ, that is, +Ÿ, Í 9Å Ð+Ñ ª 9Å Ð,Ñ In loose terms, Theorem 1.11 tells us that every poset T can be represented as a family of subsets of some set, under set inclusion. This is the first of several representation theorems for describing abstract order structures in terms of more familiar order structures. (One of the most famous representation theorems is the Cayley representation theorem of group theory, which says that every group can be represented as a subgroup of the group of permutations of a set.) Partially Ordered Sets 15 We will also have frequent use for the following generalization of the down map. Definition Let F be a nonempty subset of a poset T . 1) For + − T , we write FÆ Ð+Ñ ³ F ∩ Æ + œ Ö, − F ± , Ÿ +× read “F -down of +.” The F -down map 9FßÆ À T Ä kÐFÑ is defined by 9FßÆ Ð+Ñ œ FÆ Ð+Ñ 2) Dually, the F -up of + − T is FÅ Ð+Ñ ³ F ∩ Å + œ Ö, − F ± , +× and the corresponding F -up map 9FßÅ À T Ä kÐFÑ is defined by 9FßÅ Ð+Ñ œ FÅ Ð+Ñ We observe that the F -down map is order preserving. Height and Graded Posets In a poset T with no infinite chains, we can define the distance .Ð+ß ,Ñ from + to , , where + Ÿ , are elements of T . Definition Let T be a poset with no infinite chains. 1) If + Ÿ , , then the distance .Ð+ß ,Ñ from + to , is the maximum length of all chains from + to , , if this maximum is finite. Otherwise, the distance is infinite. 2) If T has !, then the height of + − T is the distance .Ð!ß +Ñ. (Note that . is not a distance function in the sense of a metric on a set, since it is not symmetric.) It is clear that the distance .Ð+ß ,Ñ is equal to the maximum length among all maximal chains from + to , . Of course, not all maximal chains from + to , need have the same length, which prompts the following definition. Definition The Jordan–Dedekind chain condition on a poset T is the condition that for every + , in T , all maximal chains from + to , have the same finite length. If a poset T has the Jordan–Dedekind chain condition, then .Ð+ß ,Ñ is the length of any maximal chain from + to , and so, for + Ÿ , Ÿ - , we have .Ð+ß ,Ñ .Ð,ß -Ñ œ .Ð+ß -Ñ 16 Lattices and Ordered Sets and in particular, 2Ð,Ñ œ 2Ð+Ñ .Ð+ß ,Ñ and if + ö ,, then 2Ð,Ñ œ 2Ð+Ñ " Definition A poset T is graded if there is a function 1À T Ä ™ for which 1) + , Ê 1Ð+Ñ 1Ð,Ñ 2) if + ö , then 1Ð,Ñ œ 1Ð+Ñ " We refer to such a function as a grading function. Loosely speaking, a graded poset is a poset in which we can assign a welldefined “level” to each element. We have just shown that if T has the Jordan– Dedekind chain condition, then T is graded by its height function. The converse is also true. Theorem 1.12 Let T be a poset with ! and with no infinite chains. Then T satisfies the Jordan–Dedekind chain condition if and only if T is graded by its height function. Proof. Suppose T is graded by its height function 2. If + œ -! - " - # â - 8 œ , is a maximal chain from + to , then -3 ö -3" and so 2Ð-3" Ñ œ 2Ð-3 Ñ ". Thus, 2Ð-3 Ñ œ 2Ð+Ñ 3 and so 2Ð,Ñ 2Ð+Ñ œ 8 Hence, all maximal chains from + to , have length 2Ð,Ñ 2Ð+Ñ and so T has the Jordan–Dedekind chain condition. The converse has already been established, since + , implies that .Ð+ß ,Ñ ". Chain Conditions The chain conditions are a form of finiteness condition. Definition Let T be a poset. 1) T has the ascending chain condition (ACC) if it has no infinite strictly ascending sequences, that is, for any ascending sequence :" Ÿ : # Ÿ : $ Ÿ â there is an index 8 such that :85 œ :8 for all 5 !. 2) Dually, T has the descending chain condition (DCC) if it has no infinite strictly descending sequences, that is, for any descending sequence :" : # : $ â there is an index 8 such that :85 œ :8 for all 5 !. Partially Ordered Sets 17 The following characterizations of ACC and DCC are very useful. Definition 1) A poset T has the maximal condition if every nonempty subset of T contains a maximal element. 2) Dually, a poset T has the minimal condition if every nonempty subset of T contains a minimal element. Theorem 1.13 1) A poset T has the ACC if and only if it has the maximal condition. 2) Dually, a poset T has the DCC if and only if it has the minimal condition. Proof. Suppose T has the ACC and let W © T be nonempty. Let =" − W . If =" is maximal, we are done. If not, then we can pick =# − W such that =# =" . Continuing in this way, we either arrive at a maximal element in W or we get an ascending sequence that does not become constant, which contradicts the ACC. Hence, T has the maximal condition. Conversely, if T has the maximal condition, then any ascending sequence in T has a maximal element, at which point the sequence becomes constant. Chain Conditions and Finiteness It seems intuitively clear that an infinite poset must have either an infinite chain or an infinite antichain, although a careful proof is a bit more subtle than one might think at first. It is also true that if a poset has an infinite chain, then it must have either an infinite ascending chain or an infinite descending chain. Theorem 1.14 1) An infinite poset must have one of the following: a) an infinite antichain b) an infinite strictly ascending chain c) an infinite strictly descending chain 2) A poset with no infinite chains or infinite antichains must be finite. 3) A poset with an infinite chain has an infinite strictly ascending chain or an infinite strictly descending chain. Proof. For part 1), assume that a poset T has no infinite antichains and has the ACC. We show that T fails to have the DCC. Since T has the maximal condition, it has a maximal element. Let ` œ Ö73 ± 3 − M× be the nonempty set of all maximal elements of T . Since ` is an antichain, it must be finite. Moreover, T œ -Ð Æ 73 Ñ and so one of the sets, say Æ 73" , must be infinite. Let E" œ Ö73" × and consider the poset T" œ Ð Æ 73" Ñ Ï Ö73" × of all elements of T strictly below 73" . This poset is infinite and also has no 18 Lattices and Ordered Sets infinite antichains and no infinite ascending chains. Thus, we may repeat the above argument and select an element 73# − T" such that T# œ Ð Æ 73# Ñ Ï Ö73# × is infinite. Note that 73" 73# . Continuing in this way, we get an infinite descending chain. Thus, T fails to have the DCC. Parts 2) and 3) follow from part 1). Dilworth's Theorem In 1950, Robert Dilworth proved the following very interesting result. Definition Let T be a finite poset. A chain cover of T is a collection of chains in T whose union is T . We say that a chain cover is disjoint if the chains in the cover are pairwise disjoint. If T has an antichain of size 8, then any chain cover will clearly have size at least 8, since no two elements in an antichain can be in the same chain of a chain cover. Thus, if T has width A, then all chain covers have size at least A. Dilworth proved that there always is a disjoint chain cover of size A. Theorem 1.15 (Dilworth's theorem) A finite poset has a disjoint chain cover of size A œ widthÐT Ñ. Proof. The proof is by induction on the size 8 of T . The result is true if 8 œ ". Assume it is true for posets of size less than kT k. We show that T has a chain cover of size A. The idea is to use a maximum-sized antichain E to split T into two proper subsets, to which we can apply the inductive hypothesis to get two disjoint chain covers that can be pieced together at the members of E to produce a disjoint chain cover of T . A maximum-sized antichain E œ Ö+" ß á ß +A × splits T into two subsets Å E and Æ E, that is, ÅE ∪ ÆE œ T whose intersection is E. Also, E is a maximum-sized antichain of each subset Å E and Æ E. To ensure that we choose an E for which each of these subsets is proper, we proceed as follows. Since T is finite, there exist elements 7 Ÿ Q such that Q is maximal in T and 7 is minimal in T . Let U œ T Ï Ö7ß Q × If widthÐUÑ œ 5 A, then the inductive hypothesis implies that U has a disjoint chain cover V of size 5 and so V ∪ Ö7ß Q × is a disjoint chain cover of T of size A (since it cannot have size less than A). In this case, we are done. If widthÐUÑ œ A, then a maximum-sized antichain E of U is also a maximumsized antichain of T , for which Æ E and Å E are proper in T . Let V be a disjoint Partially Ordered Sets 19 chain cover of size A for Å E and let W be a disjoint chain cover of size A for Æ E. Then for each + − E, there is a G+ − V and a H+ − W for which G+ ∩ H+ œ Ö+×. Since + is the smallest element of G+ and the largest element of H+ , it follows that G+ ∪ H+ is a chain in T . The family ÖG+ ∪ H+ ± + − E× is a disjoint chain cover of T of size A. The following important theorem on the existence of SDRs for finite families is a consequence of Dilworth's theorem. Theorem 1.16 (Philip Hall's theorem) A family Y of subsets of a finite set \ has an SDR if and only if it satisfies Hall's condition: For any " Ÿ 5 Ÿ kY k, the union of any 5 members of Y has size at least 5 , that is, kE3" ∪ â ∪ E35 k 5 Proof. Let \ œ Ö+" ß á ß +7 × and Y œ ÖE" ß á ß E8 ×. If Y has an SDR, then any 5 members of Y have an SDR and so their union must have size at least 5 . For the converse, suppose that Y satisfies Hall's condition. Consider the “membership” partial order on the set T œ \ ∪ Y defined by +3 Ÿ E4 if +3 − E4 The chains of T are the singletons and the two-element chains + E3 where + − E3 . Hence, all disjoint chain covers in T have the form (after reindexing if necessary) V œ Ö+" E" ß á ß +< E< ß +<" ß á ß +7 ß E<" ß á ß E8 × Thus, since \ is an antichain in T , a chain cover of size 7 (if it exists) must have the form V œ Ö+" E" ß á ß +8 E8 ß +8" ß á ß +7 × and so Ö+" ß á ß +8 × is an SDR for Y . Hence, it is sufficient to show that T has a chain cover of size 7, and Dilworth's theorem implies that it is sufficient to show that T has width 7. We have just noted that T has width at least 7, since \ is an antichain. Suppose that, after possible reindexing, the set T œ Ö+" ß á ß +? ß E" ß á ß E@ × is an antichain in T . We count the number of elements in W œ Ö+" ß á ß +? × ∪ E" ∪ â ∪ E@ which, of course, is at most 7. None of +" ß á ß +? lie in E" ∪ â ∪ E@ and Hall's condition implies that E" ∪ â ∪ E@ has size at least @. Hence, 20 Lattices and Ordered Sets kTk œ ? @ Ÿ kW k Ÿ 7 and so widthÐT Ñ Ÿ 7. Symmetric and Transitive Closures If a binary relation fails to be symmetric or transitive, the relation can be augmented to fix the problem. Definition Let V be a binary relation on a nonempty set \ . 1) The symmetric closure V =- of V is the smallest symmetric binary relation on \ that contains V . 2) The transitive closure V >- of V is the smallest transitive binary relation on \ that contains V . Theorem 1.17 Let V be a binary relation on a nonempty set \ . 1) The symmetric closure V =- of V is V =- œ V ∪ V " where V " œ ÖÐ+ß ,Ñ ± Ð,ß +Ñ − V×Þ 2) The transitive closure V >- of V is the relation defined as follows: Ð+ß ,Ñ − V >- if and only if there is a finite sequence +! ß á ß +8 − \ with 8 " for which +! œ +ß +8 œ , and +! V+" V+# â+8# V+8" V+8 that is, +3 V+3" for 3 œ !ß á ß 8 ". 3) a) If V is reflexive, then so is V >- . b) If V is symmetric, then so is V >c) If V is reflexive and symmetric, then V >- is an equivalence relation. d) V antisymmetric does not imply that V >- is antisymmetric. Proof. For 3b), if +V >- , , then there is a finite sequence +! ß á ß +8 − \ for which +! œ +ß +8 œ , and +! V+" V+# â+8# V+8" V+8 Since V is symmetric, we may reverse the direction of the sequence to get +8 V+8" V+8# â+# V+" V+! and so ,V >- +. For 3d), let \ œ Ö+ß ,ß -× have three distinct elements and consider the antisymmetric relation V œ ÖÐ+ß ,Ñß Ð,ß -Ñß Ð-ß +Ñ× Then V >- contains Ð+ß -Ñ as well as Ð-ß +Ñ, where + Á - . The covering relation of a poset can be characterized as follows. Partially Ordered Sets 21 Definition A binary relation V is totally intransitive if whenever +" V+# V+$ VâV+8 y 8 and +" Á +8 . for 8 $, it follows that +" V+ Theorem 1.18 A binary relation V on a set T is the covering relation ö for a partial order Ÿ on T if and only if V is totally intransitive. Proof. If ö is a covering relation on T and if +" ö +# ö +$ ö â ö +8 for 8 $, then +" +8 and so +" Á +8 . Also, +" ö y +8 . Hence, ö is totally intransitive. Conversely, suppose that V is totally intransitive. Let Ÿ be the transitive closure of the set V ∪ ÖÐ+ß +Ñ ± + − T × Then Ÿ is reflexive and transitive. To see that it is also antisymmetric, suppose that + Ÿ , and , Ÿ +. If + Á , , then +VâV, and ,VâV+ and so +VâV+, which implies that + Á +, a contradiction. Thus, Ÿ is a partial order. To see that V is the covering relation ö for Ÿ , first note that +V+ implies y . Therefore, if +V, then + Á , and + Ÿ , , which +V+V+ which implies that +V+ y ,a implies that + , . But if + - , , then total intransitivity implies that +V, contradiction. Hence, +V, implies + ö , . Also, if + ö , then + , and so +VâV, . But if +V-VâV, then + Ÿ - Ÿ , and so + ö , implies that - œ + or - œ ,, which is false. Compatible Total Orders If ÐT ß Ÿ Ñ is a poset, then a total order £ on T is compatible with the partial order Ÿ if +Ÿ, Ê +£, The problem of finding a compatible total order for a given partial order is a practical one. For example, if ÐT ß Ÿ Ñ is finite, it may be desirable to input the data T into a computer using a (linear) input order that is compatible with the partial order. For a finite poset, it is easy to find a compatible total order using a simple algorithm called topological sorting, implemented by simply taking a maximal (or minimal) element at each stage. In this way, larger (or smaller) elements are input first. We wish to prove that for any poset ÐT ß Ÿ Ñ there is a compatible total order. In fact, for a given pair of incomparable elements + and , in ÐT ß Ÿ Ñ, we can find a compatible total order £ for which + £ , . Theorem 1.19 Let ÐT ß Ÿ Ñ be a poset and let + ² , in T . The transitive closure of the relation 22 Lattices and Ordered Sets V œ Ÿ ∪ ÖÐ+ß ,Ñ× is the relation W œ Ÿ ∪ ÖÐBß CÑ ± B Ÿ +ß , Ÿ C× Moreover, W œ V >- is a partial order on T . Proof. It is clear that W is contained in V >- . Thus, if we show that W is a partial order on T , then it must be the transitive closure of V . Of course, W is reflexive since Ÿ is reflexive. As to antisymmetry, if BWC and CWB, then there are cases to consider. Let us write W w œ W Ï Ÿ œ ÖÐBß CÑ ± B Ÿ +ß , Ÿ C× 1) If B Ÿ C and C Ÿ B, then B œ C. 2) If BW wC and C Ÿ B, then B Ÿ +ß , Ÿ C and so ,ŸCŸBŸ+ which is false since + ² , . 3) If B Ÿ C and CW wB, then C Ÿ +, , Ÿ B and so ,ŸBŸCŸ+ which is false since + ² , . 4) If BW wC and CW wB, then B Ÿ +ß , Ÿ C and C Ÿ +ß , Ÿ B and so ,ŸBŸ+ which is false since + ² , . Finally, as to transitivity, suppose that BWC and CWD . Then 1) If B Ÿ C and C Ÿ B, then B Ÿ D and so BWD . 2) If BW wC and C Ÿ D , then B Ÿ +ß , Ÿ C and so B Ÿ +ß , Ÿ C Ÿ D which implies that BWD . 3) If BW wC and CW wD , then C Ÿ +ß , Ÿ D and so B Ÿ +ß , Ÿ D which implies that BWD . 4) If BW wC and CW wD , then B Ÿ +ß , Ÿ C and C Ÿ +ß , Ÿ D and so B Ÿ +ß , Ÿ D which implies that BWD . Theorem 1.20 Let ÐT ß Ÿ Ñ be a poset and let + ² , in T . Then Ÿ is contained in a total order £ on T for which + £ , . Partially Ordered Sets 23 Proof. Let f be the collection of all partial orders on T that contain Ÿ and Ð+ß ,Ñ. Then f is not empty since the transitive closure of V œ Ÿ ∪ ÖÐ+ß ,Ñ× is in f . Order f by set inclusion. Let V œ ÖT3 ± 3 − M× be a chain in f . The union T œ -T3 is a partial order on T containing Ÿ and Ð+ß ,Ñ. Thus, V has an upper bound and so by Zorn's lemma, there is a maximal order Q containing Ÿ and Ð+ß ,Ñ. Then Q must be linear, for if not, there are elements Bß C − T such that B ² C in Q and we may extend Q to a strictly larger partial order on \ containing Ÿ and ÐBß CÑ, contradicting the maximality of Q . The Poset of Partial Orders If T is a nonempty set, then the family Y of all partial orders on T is itself a partial order, where ) Ÿ . in Y if +) , Ê + ., for all +ß , − T . The smallest partial order on T is equality, since +œ, Ê +., for all partial orders . on T . We leave proof of the following as an exercise. Theorem 1.21 Let T be a nonempty set. 1) The smallest partial order on T is equality. 2) The maximal partial orders on T are the total orders. 3) If ) is a partial order on T , then ) is the intersection of all total orders on T that contain ). Exercises 1. 2. 3. 4. 5. A strict order on a nonempty set T is an irreflexive, transitive binary relation on T . Prove that there is a one-to-one correspondence between strict orders on T and partial orders on T . Let T and U be disjoint posets. Show that the union T ∪ U is a poset, with the inherited order on T and U and where : ² ; for all : − T and ; − U. Let T and U be disjoint posets. Let T ∪ U be the union with the inherited order on T and U and such that : ; for all : − T and ; − U. Prove that this is a poset. It is called the linear sum of T and U and denoted by T Š U. Find a poset T that contains an antichain E œ ÖBß Cß D× with the property that no pair of elements from E has a join but E has a join. Let E œ Ö+" ß +# ß á × and F œ Ö," ß ,# ß á × be disjoint countably infinite sets. Define a relation V on the union E ∪ F by +3" V+3 ß ,3" V,3 and ,3 V+3 24 Lattices and Ordered Sets for 3 œ "ß #ß á Þ Show that V is a covering relation. Draw a picture of the corresponding poset. 6. Let T and U be posets. a) Describe the covering relation in the product T ‚ U. b) If T and U are chains, when is T ‚ U a chain? 7. Let T be a poset. Let V+ß, be a chain from + to , and let W,ß- be a chain from , to - . a) Show that V+ß, ∪ W,ß- is a chain from + to - . b) Show that if V+ß, and W,ß- are maximal chains, then so is V+ß, ∪ W,ß- . 8. Let T be a poset. Prove that the set bÐT Ñ of down-sets is closed under arbitrary union and arbitrary intersection. 9. Let \ be a nonempty set and let T œ kÐ\Ñ. Define a relation ¥ by E ¥ F if for every + − E, there is a , − F for which + Ÿ , . Prove that if E ¥ F and F ¥ E, then E and F have the same set of maximal elements. 10. Let T and U be posets. Prove that a monotone map 0 À T Ä U is an order isomorphism if and only if 0 has a monotone inverse map 0 " À U Ä T . 11. Let T and U be finite posets and let 0 À T Ä U be a function. Prove that 0 À T Ä U is an order embedding if and only if 0 ÐBÑ 0 ÐCÑ Í BC for all Bß C − T . 12. Let T and U be posets and let 0 À T Ä U. a) Prove that 0 is monotone if and only if the induced inverse map satisfies 0 " ÐHÑ − bÐT Ñ for all H − bÐUÑ. b) Assume that 0 is monotone and let 0 " À bÐUÑ Ä bÐT Ñ be the induced inverse map from part a). Show that 0 is an order embedding if and only if 0 " is surjective. c) Assume that 0 is monotone and let 0 " À bÐUÑ Ä bÐT Ñ be the induced map from part a). Show that 0 is surjective if and only if 0 " is injective. 13. Let T be a poset. Prove that the down map is an order embedding from T into kÐT Ñ. 14. Let M8 œ Ö"ß á ß 8× and let 0 À kÐM8 Ñ Ä Ö!ß "×8 be defined by 0 I œ Ð/" ß á ß /8 Ñ where /3 œ " if and only if 3 − I . Prove that 0 is an order isomorphism. 15. Let T be a poset and let T ` be the dual poset. Prove that bÐT Ñ ¸ ÐbÐT ` ÑÑ` 16. Let T and U be posets. Prove that bÐT “ UÑ ¸ bÐT Ñ ‚ bÐUÑ. 17. Let T be a poset of size at least +, ". Prove that T contains either an antichain of size at least + " or a chain of size at least , ". 18. Let T be a finite poset. a) Show that kbÐT Ñk is equal to the number of antichains in T . Partially Ordered Sets 25 b) Show that for B − T , kbÐT Ñk œ kbÐT Ï ÖB×Ñk kbÐT Ï Ð Æ B ∪ Å BÑÑk 19. (Dual to Dilworth's theorem) An antichain cover of T is a collection of antichains in T whose union is T . Prove the dual to Dilworth's theorem: A poset T with finite height 2 has an antichain cover of size 2. 20. Let £ be a preorder on \ . Define a binary relation µ on \ by +µ, if + £ , and , £ + a) Show that µ is an equivalence relation on \ . Denote the equivalence class containing + − \ by Ò+Ó. b) Let +ß , − \ . Show that if + £ , , then B £ C for all B − Ò+Ó and C − Ò,Ó. c) Show that the binary relation on the set \Î µ of equivalence classes of \ under µ defined by Ò+Ó Ÿ Ò,Ó 21. 22. 23. 24. 25. 26. 27. if + £ , is a partial order on \Î µ . d) Given a partial order Ÿ on T , let £ be defined on the power set kÐT Ñ by W £ X if for each = − W , there is a > − X for which = Ÿ >. Prove that £ is a quasiorder. Let ÐT ß Ÿ Ñ be a poset and let + Ÿ y , in T . Show that Ÿ is contained in a total order £ on T for which , £ +. Let T be a nonempty set. Prove the following: a) The smallest partial order on T is equality. b) The maximal partial orders on T are the total orders. c) (Szpilrajn, 1930) If ) is a partial order on T , then ) is the intersection of all total orders on T that contain ). Let \ be a topological space. Let clÐEÑ denote the closure of E in \ . a) Show that B Ÿ C if B − clÐÖC×Ñ is a preorder on \ . b) Show that Ÿ is a partial order if and only if \ is X! , that is, given any B Á C − \ there is an open set containing one of B or C but not both. The linear kernel OÐT Ñ of a poset T is the set of all elements of T that are comparable to every element of T . Prove that OÐT Ñ is the intersection of all maximal chains of T . Prove that Zorn's lemma is equivalent to the Hausdorff maximality principle. Prove that the Hausdorff maximality principle implies Tukey's lemma. Prove that Tukey's lemma implies the axiom of choice. Chapter 2 Well-Ordered Sets This chapter is an introduction to the subject of well-ordered sets and ordinal and cardinal numbers. While this material is more typically found in a textbook on set theory, we will have a few occasions to use the basic theory of ordinals, most notably in describing conditions that characterize the incompleteness of a lattice and in the discussion of fixed points. However, the chapter may be omitted upon first reading without severe consequences. Well-Ordered Sets We recall some earlier facts. Definition A poset Ð[ ß Ÿ Ñ is a well-ordered set if every nonempty subset of [ contains a least element. In this case, the order Ÿ is called a well-ordering of [ . A nonempty set \ for which a well-ordering exists is called a wellorderable set. A well-ordered set is totally ordered, since any doubleton set Ö+ß ,× has a least element. Also, every nonmaximal element + − [ has a unique cover, since the set Y+ œ ÖB − [ ± B +× is nonempty and so has a least element D , which covers +. Moreover, if + ö C, then C − Y+ and so + D Ÿ C. Hence, C œ D . Initial Segments In a well-ordered set, the following down-sets are special. Definition Let T be a poset. For + − T , the set T Ò+Ó ³ ÖB − T ± B +× is called the initial segment of T at +. (Note that +  T Ò+Ó.) We denote the S. Roman (ed.), Lattices and Ordered Sets, doi: 10.1007/978-0-387-78901-9_2, © Steven Roman 2008 27 28 Lattices and Ordered Sets family of initial segments of T by segÐT Ñ. When we write T Ò+Ó, it is with the tacit understanding that + − T . If T is totally ordered, then +Ÿ, Í T Ò+Ó © T Ò,Ó and so the initial segment map MÀ T Ä segÐT Ñ defined by MÐ+Ñ œ T Ò+Ó is an order isomorphism. It follows that segÐT Ñ ¸ T and so segÐT Ñ is totally ordered by set inclusion. We can say more when T is well-ordered. Theorem 2.1 Let [ be a well-ordered set. 1) The initial segments in [ are precisely the proper down-sets of [ , that is, segÐ[ Ñ œ b ‡ Ð[ Ñ ³ bÐ[ Ñ Ï Ö[ × 2) The function MÀ [ ¸ b‡ Ð[ Ñ defined by MÐ+Ñ œ [ Ò+Ó is an order isomorphism and so bÐ[ Ñ is well-ordered by set inclusion. Proof. For part 1), the sets [ Ò+Ó are clearly proper down-sets. If H is a proper down-set of [ , then [ Ï H has a least element + and H œ [ Ò+Ó. To see this, it is clear that [ Ò+Ó © H, since + is the least element not in H. Also, if . − H but . +, then + Ÿ . − H and since H is a down-set, we get + − H, which is false. For part 2), if + Ÿ , , then certainly [ Ò+Ó © [ Ò,Ó. Conversely, if [ Ò+Ó © [ Ò,Ó but , +, then , − [ Ò+Ó © [ Ò,Ó and so , − [ Ò,Ó, which is false. Hence, + Ÿ , and so M is an order embedding Finally, M is surjective by part 1) and so M is an order isomorphism. Isomorphisms of Well-Ordered Sets We will have use for the following concept. Definition Let T be a poset. A function 0 À T Ä T is inflationary if B Ÿ 0 ÐBÑ for all B − T . The term increasing is also used for inflationary functions but unfortunately, some authors use increasing to mean monotone, so we have decided to avoid the term altogether. Strictly monotone functions need not be inflationary. However, the situation is different for well-ordered sets and this has some important consequences. Theorem 2.2 Let [ be a well-ordered set. Well-Ordered Sets 29 1) A strictly monotone function 0 À [ Ä [ is inflationary. 2) [ is not isomorphic to any subset of any initial segment [ , nor are any two distinct initial segments of [ isomorphic. 3) There is at most one isomorphism between any two well-ordered sets. In particular, the identity is the only automorphism of [ . Proof. For part 1), if there is an B − [ such that 0 ÐBÑ B, then the set W œ ÖB − [ ± 0 ÐBÑ B× is nonempty and so has a least element +. But applying 0 to the inequality 0 Ð+Ñ + gives 0 Ð0 Ð+ÑÑ 0 Ð+Ñ which shows that 0 Ð+Ñ − W , a contradiction to the minimality of +. For part 2), if 0 À [ ¸ W © [ Ò+Ó, then since 0 is inflationary, we have + Ÿ 0 Ð+Ñ, which is impossible since, then 0 Ð+Ñ Â [ Ò+Ó. Also, if + , , then [ Ò+Ó is an initial segment of the well-ordered set [ Ò,Ó and so is not isomorphic to [ Ò,Ó. For part 3), if 0 À [ ¸ [ is an automorphism, then 0 and 0 " are inflationary and so B Ÿ 0 ÐBÑ and B Ÿ 0 " ÐBÑ. Applying 0 to the latter gives 0 ÐBÑ Ÿ B and so 0 ÐBÑ œ B, that is, 0 is the identity map. Hence, if 0 ß 1À [" ¸ [# , then 1" 0 is an automorphism of [" and so 1" 0 is the identity, whence 0 œ 1. The fact that a strictly monotone function 0 À [ Ä [ is inflationary has further important consequences. In particular, the following result about pasting together isomorphisms is very useful. For instance, it shows immediately that if [" and [# are nonisomorphic well-ordered sets, then one is isomorphic to an initial segment of the other. Theorem 2.3 Let [" and [# be well-ordered sets. 1) The family T œ Ö0 À H ¸ I ± H − bÐ[" Ñß I − bÐ[# Ñ× is nonempty. Also: a) If 0 À H Ä I and 1À Hw Ä I w are in T and H © Hw then 1lH œ 0 . b) For each H − bÐ[" Ñ, there is at most one 0 − T with domain H and so we may totally order T by Ð0 À H ¸ IÑ Ÿ Ð1À Hw ¸ I w Ñ iff H © Hw 30 Lattices and Ordered Sets 2) If E³ . H and F³ HœdomÐ0 Ñ where 0 −T . I IœimÐ0 Ñ where 0 −T then the function J À E ¸ F defined by J ÐBÑ œ 0 ÐBÑ for any 0 with B − domÐ0 Ñ is the largest element of T . Moreover, either E œ [" or F œ [# . 3) (Cantor, 1897) Exactly one of the following holds: a) [" ¸ [ # b) [" ¸ [# Ò+Ó for some + − [# c) [# ¸ [" Ò,Ó for some , − [" Moreover, the isomorphism in question is unique. Proof. For part 1), the family T is nonempty since the function that takes the smallest element of [" to the smallest element of [# is in T . Next, we observe that there is at most one function in T for each H − bÐ[" Ñ, for if 0À H ¸ I and 1À H ¸ I w then I ¸ I w are isomorphic down-sets in [# and so Theorem 2.2 implies that I œ I w . Then 0 ß 1À H ¸ I and so 0 œ 1. Finally, if 0À H ¸ I and 1À Hw ¸ I w are in T and H © Hw , then 1lH À H ¸ 1ÐHÑ is in T and so 1lH œ 0 . For part 2), the function J is well defined, since B − domÐ0 Ñ ∩ domÐ1Ñ implies that 0 ÐBÑ œ 1ÐBÑ and so the definition of J ÐBÑ is independent of the choice of 0 . Since B Ÿ C in E Í 0 ÐBÑ Ÿ 0 ÐCÑ for some 0 − T Í J ÐBÑ Ÿ J ÐCÑ and since J is clearly surjective, it is an order isomorphism. Since E and F are down-sets, it follows that J − T . Clearly, J is the largest element of T . Finally, if E œ [" Ò+Ó and F œ [# Ò,Ó, then we can extend J À [" Ò+Ó ¸ [# Ò,Ó to an isomorphism J À Æ + ¸ Æ , of down-sets by setting J Ð+Ñ œ , . This contradicts the fact that J is the largest element of T . Hence, either E œ [" or F œ [# . For part 3), Theorem 2.2 shows that at most one of 3a)–3c) can hold and that the isomorphism must be unique. But the isomorphism J establishes that one of 3a)–3c) must hold. Well-Ordered Sets 31 Ordinal Numbers The ordered set \ œ gß Ög×ß egß Ög×fß egß Ög×ß egß Ög×ff has the property that each element is the set of all elements that came before it. Thus, every element of \ is also a subset of \ . There is a name for such sets. Definition A set \ is transitive if every element of \ is a subset of \ , that is, if B−+−\ Ê B−\ Note that to say that a set is transitive is not the same as saying that membership is a transitive relation on \ , that is, for any Bß Cß D − \ , B−C−D Ê B−D This is two different uses of the term transitive: one as it relates to a set and one as it relates to a binary relation. However, both senses are used in the following definition of the principle object of study of this chapter. Definition A set α is an ordinal number or ordinal if the following hold: 1) α is transitive, that is, +−,−α Ê +−α 2) Membership is a strict (asymmetric and transitive) well-ordering of α. We will reserve lowercase Greek letters for ordinal numbers and denote the fact that α is an ordinal by α − ord. Somewhat as an aside, the family ord of all ordinals is too large to be considered a set, since doing so leads to certain paradoxes of set theory. To avoid these paradoxes, set theorists have defined classes. Thus, ord is a class. Much, but not all, of set theory can be applied to classes. For more on this, we refer the reader to a book on set theory. Thus, for elements of an ordinal number α, the phrase “less than” means “element of.” Let us make a few observations that follow directly from the definition. Let α − ord. 1) Since membership is a strict order on α, it follows that αÂα which implies that the elements of α are proper subsets of α. 2) Consider the chain +−,−-−α where α − ord. The transitivity of the set α implies that , − α and so 32 Lattices and Ordered Sets + − , − α, whence + − α. More generally, +" − + # − â − + 8 − α Ê +3 − α for all 3 œ "ß á ß 8 Also, the transitivity of the membership relation gives +" − + # − â − + 8 − α Ê +3 − +4 for all 3 4 3) (The elements of α are the initial segments of α) The initial segment αÒ+Ó of α is by definition the set of all elements of α that are elements of +, that is, αÒ+Ó œ + In words, each element + − α is its own initial segment. 4) (The elements of α are ordinals) Any element + − α is also an ordinal. In fact, the transitivity of membership in α shows that + is a transitive set, for if B − C − +, then B − + and so C © +. Also, since + © α, it follows that membership is a strict well-ordering on +. 5) (No other proper subsets of α are ordinals) A proper subset W § α is an ordinal if and only if W is an initial segment (element of) α. For if W − ord, then W is a proper down-set in α, since if = − W and + − = implies that + − W . Hence, Theorem 2.1 implies that W is an initial segment of α. 6) (In ord, proper inclusion is membership) If αß " − ord, then α§" Í α œ " Ò+Ó for some + − " Í α−" Thus, for ordinals, membership is equivalent to proper inclusion. A word on notation: It is customary to use the symbol as an alternative notation for membership. Thus, if αß " − ord, then α" means α − " The symbol reminds us more of a strict order, but the symbol − is more meaningful in this context. However, there is no commonly accepted notation involving − to express the fact that α − " or α œ " and so we use α Ÿ " . The Natural Numbers The prototypical examples of ordinals are the natural numbers. Indeed, the natural number ! is defined to be the empty set, which is an ordinal. The other natural numbers are defined, in their natural order, by specifying that the natural number 8 is the set of all natural numbers that have already been defined. Thus, Well-Ordered Sets 33 !œg " œ Ö!× œ Ög× # œ Ö!ß "× œ egß Ög×f $ œ Ö!ß "ß #× œ egß Ög×ß egß Ög×ff % œ Ö!ß "ß #ß $× œ egß Ög×ß egß Ög×fß egß Ög×ß egß Ög×fff ã 8 œ Ö!ß "Þß á ß 8 "× œ 8 " ∪ Ö8 "× ã By definition, an ordinal is finite if it is a natural number; otherwise, it is infinite. The set , under the usual order !"#â is an ordinal number, customarily denoted by = when thought of as an ordinal. (We leave proof to the reader.) Also, = is the smallest infinite ordinal, in the sense that any proper subset of = that is also an ordinal must be a natural number. For if α is an ordinal and α § =, then α œ =Ò8Ó œ 8 for some 8 − . Characterizing Ordinals by Initial Segments We have seen that the initial segments of α − ord coincide with the elements of α, that is, αÒ+Ó œ + for all + − α. Conversely, suppose that Ð[ ß Ñ is a well-ordered set for which [ Ò+Ó œ + for all + − [ . Then [ is clearly transitive and the strict well-ordering must be the membership relation, since +−, Í + − [ Ò,Ó Í +, Hence, [ is an ordinal. This gives a very useful characterization of ordinals which is sometimes taken as the definition of ordinal number. Theorem 2.4 A well-ordered set Ð[ ß Ñ, where is a strict well-ordering, is an ordinal if and only if + œ [ Ò+Ó for all + − [ . In this case, the strict order relation is membership, that is, + , in [ Í +−, 34 Lattices and Ordered Sets The Successor of an Ordinal The successor of a set \ is the set succÐ\Ñ œ \ ∪ Ö\× Note that the natural number 8 " is defined to be the successor of the natural number 8. Theorem 2.5 If α is an ordinal, then so is its successor. Proof. The set succÐαÑ œ α ∪ Öα× is transitive, since if B − + − α ∪ Öα×, then either + œ α, in which case B − α or else B − + − α, in which case B − α. Also, α is the largest element in succÐαÑ and so membership is a strict well-ordering of succÐαÑ. Sets of Ordinals We have seen that membership, which is the same as strict inclusion, is a strict order on the class ord. It follows, of course, that membership is a strict order on any nonempty set W of ordinals. But we can say more, namely, the union -W and intersection +W of W are also ordinals and the intersection - is actually the least element of W and the union is the least upper bound for W . It follows that any nonempty set W of ordinals is well-ordered by membership (strict inclusion). To see this, note first that the union or intersection of transitive sets is transitive. Also, since +W and -W are both sets of ordinals, membership is a strict order on these sets. Since the intersection - œ +W is a subset of any ordinal α − W , it follows that - is well-ordered by membership and so - is an ordinal. Moreover, since - © α for all α − W , it follows that - œ α or - − α for every α − W . Thus, - is a lower bound for W . Also, if - − α for all α − W , then - − +W œ -, which is false. Hence, - œ α for some α − W and so - − W is the least element of W . Finally, since every nonempty set of ordinals has a least element, it follows that the union -W is well-ordered by membership and is therefore an ordinal. Theorem 2.6 Let W be a nonempty set of ordinals, strictly ordered by membership. 1) +W is an ordinal and is the least element of W . a) Therefore, every nonempty set of ordinals is well-ordered by membership. In particular, the trichotomy law holds, that is, if αß " − ord, then exactly one of the following holds: α − "ß αœ" or " − α b) A set of ordinals is an ordinal if and only if it is transitive. 2) -W is an ordinal and is the least upper bound (supremum) of W . Well-Ordered Sets 35 a) Hence, for every set W of ordinals, there is an ordinal that is greater than every ordinal in W , namely, succÐ-WÑ. 3) The set of finite ordinals is the set of natural numbers. Proof. For part 3), the definition of natural number implies that all natural numbers are finite ordinals. Conversely, if α − ord is finite, then the trichotomy law implies that α − = and so α œ =Ò8Ó œ 8, for some 8 − . Ordinals Are Order Types We can now show that every well-ordered set is isomorphic to exactly one ordinal. Theorem 2.7 Every well-ordered set [ is isomorphic to a unique ordinal number, which is called the order type of [ . Proof. Since if αß " − ord, then one of α or " is an initial segment of the other, it follows that no two distinct ordinals are order isomorphic and this proves the uniqueness part of the theorem. For the existence part, if there is an ordinal α for which [ ¸ α or [ ¸ αÒ+Ó for some + − α, then we are done, since αÒ+Ó is also an ordinal. So let us assume that this is not the case. Then Theorem 2.3 implies that for every α − ord, there is a necessarily unique + − [ for which α ¸ [ Ò+Ó. It follows from an axiom schema of set theory that ord is a set, which is false, since for any set W of ordinals, there is an ordinal that is strictly greater than all elements in W . (The axiom schema to which we refer is the Axiom Schema of Replacement. We refer the interested reader to a book on set theory for more details.) Successor and Limit Ordinals The successor of an ordinal α is denoted by α ", that is, α " ³ succÐαÑ œ α ∪ Öα× An ordinal " of the form " œ α " is called a successor ordinal. A nonsuccessor ordinal is called a limit ordinal. Note that αŸ" Í α"" We leave proof of the following as an exercise. Theorem 2.8 Let αß " − ord. 1) If α − " , then either α " − " or α " œ " . 2) α " is the unique cover of α. 3) The following are equivalent for an ordinal -: a) - is a limit ordinal b) α − ord, α − - Ê α " − - 36 Lattices and Ordered Sets c) - is the union of all ordinals contained in -, that is, - œ .α α−- Transfinite Sequences The notion of sequence can be extended into the transfinite. Definition 1) Let E be a set and let $ − ord. A function 0 À $ Ä E is called a transfinite sequence of length $ in E and is often denoted by f œ Ø+α ± α $ Ù where αα œ 0 ÐαÑ. The set imÐf Ñ œ Ö+α ± α $ × is called the underlying set, range or image of f and $ is called the index ordinal of f . 2) By analogy, a “function” 0 À ord Ä A is called a sequence on ord and is denoted by Ø+α ± α − ordÙ We will say that an element + is in a sequence or belongs to a sequence f if + is in the underlying set of f . The image Ö+α ± α $ × of an injective transfinite sequence Ø+α ± α $ Ù is well-ordered by the index ordinal $ : +α Ÿ + " if α−" Conversely, if \ is a well-ordered set, then it is isomorphic to a unique ordinal $ and the unique isomorphism 0 À $ ¸ \ is a transfinite sequence. Thus, an injective transfinite sequence with range \ is essentially just a way to wellorder the set \ . Subsequences Suppose that 0 À $ Ä E is a transfinite sequence. If " Ÿ $ and if 1À " Ä $ is a strictly increasing function, then the composition 0 ‰ 1À " Ä E is called a subsequence of 0 . A subsequence of a sequence 0 is just a way of selecting a portion of the image of 0 , while at the same time preserving the original order that 0 imparts to its image. We can take a slightly different viewpoint of subsequences as follows. Suppose that 0 À $ Ä E is a transfinite sequence and [ © $ is a nonempty subset of $ , with the well-order inherited from $ . Then there is a unique order isomorphism 1À α ¸ [ , where α − ord. The map Ð0 ‰ 1ÑÀ α Ä E is a subsequence of 0 , but it will sometimes be convenient to think of the map 0 l[ À [ Ä E as a subsequence of 0 , which does no harm since [ uniquely determines α and 1. As mentioned, the notation f œ Ø+α ± α $ Ù is often used to denote a transfinite sequence 0 À $ Ä E, where 0 ÐαÑ œ +α . Then a subsequence Well-Ordered Sets 37 Ð0 ‰ 1ÑÀ " Ä E of 0 is denoted by g œ Ø+1ÐαÑ ± α " Ù. Thus, to say that an element +- in f is in g is to say that +- œ +1ÐαÑ for some α − " . Transfinite Induction The well-known principle of induction for the natural numbers can be generalized as follows. Theorem 2.9 (Transfinite Induction: Version 1) Let T ÐBÑ be a property of ordinals, that is, for each α − ord, the statement T ÐαÑ is either true or false. If the following inductive hypothesis holds: T Ð" Ñ holds for all " α Ê T ÐαÑ holds then T ÐαÑ holds for all ordinals α. Proof. If T ÐαÑ does not hold for some α − ord, then let \ be the nonempty set of all ordinals " − α for which T Ð" Ñ fails and let - be the least element of \ . Thus, T Ð-Ñ fails, but T Ð" Ñ holds for every " − -, contradicting the inductive hypothesis. There is another common version of transfinite induction. Theorem 2.10 (Transfinite Induction: Version 2) Let T ÐBÑ be a property of ordinals, that is, for each α − ord, the statement T ÐαÑ is either true or false. Assume that 1) T Ð!Ñ holds 2) If α is any ordinal, then T ÐαÑ holds Ê T Ðα "Ñ holds 3) If α is a limit ordinal, then T Ð" Ñ holds for all " α Ê T ÐαÑ holds Then T ÐαÑ holds for all ordinals. Proof. If T ÐαÑ does not hold for some α − ord, then let \ be the nonempty set of all ordinals " α for which T Ð" Ñ fails and let - be the least element of \ . Thus, T Ð-Ñ fails but T Ð" Ñ holds for every " -. If - œ !, then we are in violation of 1); if - œ α " is a successor ordinal, then we are in violation of 2) and if - is a limit ordinal, then we are in violation of 3). Cardinal Numbers Cardinal numbers are special types of ordinal numbers. Definition Two sets \ and ] are equipotent (also called equipollent or equinumerous) if there is a bijection between \ and ] . 38 Lattices and Ordered Sets We use the notation \ ä ] to denote the fact that there is an injection from \ to ] and the notation \ Ç ] to denote the fact that there is a bijection from \ onto ] , that is, that \ and ] are equipotent. Definition An ordinal α is an initial ordinal or a cardinal if α is not equipotent to any smaller ordinal " − α. We denote the class of all cardinal numbers by Cn. The natural numbers are initial ordinals, as is the ordinal =. Also, it is not hard to show that any infinite cardinal number " is a limit ordinal. The Cardinality of a Set In general, a well-orderable set \ may have many well-orderings. For each well-ordering Ÿ on \ , the well-ordered set Ð\ß Ÿ Ñ has a unique order type "Ÿ . The family of well-orderings on \ is a set and therefore so is the set OrdTypesÐ\Ñ of all order types corresponding to the well-orderings of \ . Since an order isomorphism is also a bijection, each order type is equipotent to \ and so OrdTypesÐ\Ñ © Ö" − ord ± " is equipotent to \× On the other hand, if α − ord is equipotent to \ , then any bijection 0 À α Ç \ can be used to transfer the well-ordering from α to \ , making Ð\ß Ÿ Ñ a wellordered set. Then 0 is an order isomorphism and so α − OrdTypesÐ\Ñ. Thus, OrdTypesÐ\Ñ œ Ö" − ord ± " is equipotent to \× Therefore, the least member α of OrdTypesÐ\Ñ is an initial ordinal, since # − α implies that # is not equipotent to \ and therefore is not equipotent to α. Theorem 2.11 Let \ be a well-orderable set. Then OrdTypesÐ\Ñ œ Ö" − ord ± " is equipotent to \× and the smallest order type for \ is an initial ordinal, called the cardinality of \ and denoted by k\ k or cardÐ\Ñ. Example 2.12 Consider the set of natural numbers. When is given the usual order !"#â then the resulting well-ordered set is denoted by =. We may also well-order by giving the positive integers the usual order and declaring that ! is the largest element, that is, "#â and 8 ! for all positive integers 8 Let us denote this well-ordered set by [ . Then the map Well-Ordered Sets 39 0 À [ Ä = " œ = ∪ Ö=× defined by 0 Ð5Ñ œ œ for 5 ! for 5 œ ! 5" = is an order isomorphism and so [ ¸ = ". In a similar way, we can well-order so that [ ¸ = 5 for 5 ". However, = is the smallest infinite ordinal and so it is the least ordinal that is equipotent to , that is, kk œ =. In this book, we will assume the axiom of choice and its equivalents (unless otherwise noted). In particular, we assume that any set is well-orderable. Hence, any set has a cardinal number, as we have defined the concept. (If one does not assume that every set is well-orderable, then one can define a more general notion of cardinal number or simply restrict the definition to well-orderable sets.) The Cantor–Schröder–Bernstein Theorem The well known Cantor–Schröder–Bernstein theorem says that \ä] and ] ä\ Ê \Ç] This theorem is usually proved by actually constructing a bijection between \ and ] . However, with our background in ordinal numbers, we can also prove the theorem by proving that \ä] Í k\ k Ÿ k] k since the theorem, then follows from the trichotomy law k\ k Ÿ k] k and k] k Ÿ k\ k It is easy to see that \Ç] Í Ê k\ k œ k] k k\ k œ k] k since \ Ç k\ k for all sets \ and no two distinct cardinal numbers are equipotent. Now suppose that 0 À \ ä ] . If we well-order ] to have order type " œ k] k, then 0 \ inherits this well-ordering and so has an order type α. Moreover, α ¸ 0\ © ] ¸ " and so α is order isomorphic to a subset of " . This implies that " cannot be an initial segment of α and so α Ÿ " . Since α is an order type of \ , we have k\ k Ÿ α Ÿ " œ k] k 40 Lattices and Ordered Sets For the converse, if \ Ÿ ] , then k\ k © k] k and there are maps \ Ç k\ k ä k] k Ç ] where the center map is inclusion. The composition is an injection from \ into ] . Thus, \ä] Í k\ k Ÿ k] k Incidentally, this justifies the introduction of the expression k\ k Ÿ k] k to mean that \ ä ] , which one often finds in books that introduce the concept of the “cardinality” of a set before introducing cardinal numbers. Theorem 2.13 If ] and \ are sets then 1) k\ k Ÿ k] k if and only if \ ä ] 2) k\ k œ k] k if and only if \ Ç ] 3) (Cantor–Schröder–Bernstein theorem) \ä] and ] ä\ Ê \Ç] Cantor's Theorem Another famous theorem associated with Cantor shows that there are arbitrarily large cardinal numbers. Theorem 2.14 (Cantor's theorem) If \ is a set then k\ k kkÐ\Ñk. Proof. Let 0 À \ Ä kÐ\Ñ be defined by 0 ÐBÑ œ ÖB×. This is an injection, showing that k\ k Ÿ kkÐ\Ñk. But if k\ k œ kkÐ\Ñk, then there is a bijection 1À \ Ä kÐ\Ñ. Let W œ ÖB − \ ± B  1ÐBÑ× The surjectivity of 1 implies that W œ 1Ð=Ñ for some = − \ . But = − W implies that =  1Ð=Ñ œ W and =  W implies that = − 1Ð=Ñ œ W , both a contradiction. Hence 1 cannot exist and so k\ k œ kkÐ\Ñk cannot hold. We mention without proof that the cardinality of the set k! Ð\Ñ of all finite subsets of an infinite set \ is the same as the cardinality of \ . The Alephs Cantor's theorem implies that the set ÖC − Cn ± D C Ÿ kkÐDÑk× is nonempty and so has a least element. This least element is the smallest cardinal greater than D. Definition If D is a cardinal, then the smallest cardinal 2ÐDÑ greater than D is called the Hartogs number of D. Well-Ordered Sets 41 The Hartogs number and transfinite recursion (which we will not discuss formally in this brief introduction) can be used to define a strictly increasing transfinite sequence of cardinal numbers, indexed by ord. Definition Define a strictly increasing transfinite sequence Øiα ± α − ordÙ as follows: 1 ) i! œ = 2) If α " is a successor ordinal, then iα" œ 2Ðiα Ñ 3) If - is a limit ordinal, then i- œ supÖi" ± " -× œ .Öi" ± " -× The numbers iα are called alephs. Theorem 2.15 The alephs are precisely the infinite cardinal numbers. Proof. Since iα 2Ðiα Ñ and since, for any limit ordinal -, i" i" " Ÿ ifor any " -, it follows that the sequence of alephs is strictly increasing. Moreover, since i! is an infinite set, all alephs are infinite cardinals. For the converse, it is easy to see by transfinite induction on α that all cardinals D less than iα are alephs. If α œ !, then the result holds vacuously. If the result holds for α and D iα" , then D Ÿ iα . But if D œ iα , then D is an aleph and if D iα , then the inductive hypothesis implies that D is an aleph. Finally, if is a limit ordinal, then D i- implies that D i" for some " - and the inductive hypothesis implies that D is an aleph. Thus, it is sufficient to show that any cardinal D is less than some aleph. For this, we show by transfinite induction that any ordinal α satisfies α Ÿ iα , since then D Ÿ iD iD" . This is clear for α œ !. Also, iα 2Ðiα Ñ gives α Ÿ iα Ê α " Ÿ iα " Ÿ 2Ðiα Ñ œ iα" Finally, if - is a limit ordinal and α iα for all α - , then - œ . α Ÿ . iα œ i α−- α−- The question of how big a “jump” occurs in going from the cardinality of a set \ and the cardinality of its power set kÐ\Ñ is a very famous one. The continuum hypothesis is the statement that the cardinality of the power set kÐÑ, which is the same as the cardinality of the real numbers ‘, is equal to i" . Kurt Gödel showed in 1940 that the continuum hypothesis is consistent with, 42 Lattices and Ordered Sets that is, cannot be disproven from, the usual Zermelo–Fraenkel axioms of set theory, with or without the axiom of choice. In 1963, Paul Cohen showed that the continuum hypothesis is independent of, that is, cannot be proven from, the Zermelo–Fraenkel axioms, with the axiom of choice. Hence, the continuum hypothesis is independent of the Zermel–Fraenkel axioms and the axiom of choice. The generalized continuum hypothesis is the statement that #iα œ iα" for all ordinals α. Ordinal and Cardinal Arithmetic The arithmetic of ordinal numbers is defined as follows. Definition (Ordinal arithmetic) Addition: 1) α ! œ α 2) α " œ succÐαÑ 3) α Ð" "Ñ œ Ðα " Ñ " for all " − ord 4) α - œ -" −- Öα " × for all limit ordinals -. Multiplication: 5) α † ! œ ! 6) α † Ð" "Ñ œ α † " α for all " − ord 7) α † - œ -" −- Öα † " × for all limit ordinals -. Exponentiation: 8) α ! œ " 9) α" " œ α" † α for all " − ord 10) α- œ -" −- Öα" × for all limit ordinals -. Here is a short list of the first few infinite ordinals: =ß = "ß = #ß á ß = = œ = † #ß = † # "ß = † # #ß á ß = † # = œ = † $ß ã = † 5 "ß á ß = † Ð5 "Ñ ã The next limit ordinal is =# œ = † = œ -5 = † 5 Now we begin again, replacing each = above by =# and continuing: =# ß á ß =# † = œ =$ ß =$ "ß á ß =5 ß á ß == œ -5 =5 This process continues: Well-Ordered Sets 43 = == "ß á ß =Ð= Ñ ß á Note that =†#Á#†= since the latter is # † = œ . Ð# † 5Ñ œ = 5− Thus, ordinal multiplication is not commutative. Since cardinal numbers are ordinal numbers, we already have a definition of sum, product and exponential for cardinal numbers. However, as measures of the size of sets regardless of any order structure, there is a more appropriate set of operations, called cardinal sum, cardinal product and cardinal exponentiation. These operations are defined in terms of set operations. Definition (Cardinal arithmetic) 1) iα i" œ ki+ “ i" k, where “ denotes the disjoint union. 2) iα † i" œ ki+ ‚ i" k, where ‚ denotes the cartesian product. i 3) iα" œ kÖ0 À i" Ä iα ×k. We omit the proof of the following theorem. Theorem 2.16 For infinite cardinal numbers D and C, we have D C œ D † C œ maxÖDß C× Also, if k! Ð\Ñ is the set of all finite subsets of an infinite set \ then kk! Ð\Ñk œ k\ k Complete Posets It is well known that the union of an arbitrary family of subgroups of a group K need not be a subgroup of K. However, the union of a chain of subgroups of K is a subgroup of K. More generally, the union of a directed family of subgroups of K is a subgroup of K. Similar statements can be made in the context of other algebraic structures, such as vector spaces or modules. Actually, the statement about directed sets is not more general than the statement about chains. We will prove that if every chain in a poset T has a join, then every directed set also has a join. 44 Lattices and Ordered Sets Definition Let T be a poset. 1) T is chain-complete or inductive if every chain in T , including the empty chain, has a join. Thus, T has a smallest element. 2) T is complete or a CPO if T has a smallest element and if every directed subset of T has a join. We should mention that the adjective complete has a different meaning when applied to posets than when applied to lattices. (We will define complete lattices in a later chapter.) The proof that chain-completeness implies completeness uses the fact that any infinite directed set H is the union of a strictly increasing transfinite sequence of directed subsets of H of cardinality less than that of H. Our proof of this result follows the lines of Markowsky [43], which is a sharpening of Iwamura [35]; see also Maeda [41], Skornyakov [57] and Mayer-Kalkschmidt [44]. Theorem 2.17 An infinite directed poset H is the union of a strictly increasing transfinite sequence f œ ØHα ± α kHkÙ of directed subsets Hα for which kHα k kHk. In particular, if α is finite, then kHα k is finite and if α is infinite, then kHα k Ÿ kαk. Proof. Well-order H using the cardinal D œ kHk as index set, that is, H œ Ø.α ± α DÙ We will define the sequence f in such a way that for all α Dß Ö.% ± % α× © Hα Then for any " D, we have ." − H" " © -Hα and so -Hα œ H. (2.18) The finite part of the sequence is easy to define: At each step, just include the smallest element of H (under the well-ordering) not yet in the sequence, along with an upper bound (under the partial order) for what has come before. Specifically, let H! œ Ö.! × and let H5" œ H5 ∪ Ö/5" ß ?5" × where /5" is the least element not in H5 and ?5" is any upper bound for the finite set H5 ∪ Ö/5" ×. It is clear that ØH5 ± 5 − Ù is a strictly increasing sequence of finite directed (in fact, bounded above) sets. Moreover, Ö.! ß á ß .5 × © H5 It follows that if D œ =, then -H5 œ H and the proof is complete. So assume that D =. Well-Ordered Sets 45 Assume that for " D, we have constructed a strictly increasing sequence ØHα ± α " Ù of directed sets Hα for which kHα k Ÿ kαk and (2.18) holds for all α " . This has been done for " œ =. Construction of the next element H" in the sequence depends on the type of the index " . For a limit ordinal " , we accumulate all that came before: H" œ . Hα α" Then H" is directed and α " implies that Hα § Hα" © H" . Also, kH" k Ÿ α" kHα k Ÿ k" k † k" k œ k" k Moreover, (2.18) holds since if % " , then % " " and so .% − H%" © H" If " œ $ " is an infinite successor ordinal, we define H$" in a manner that is similar in spirit to the finite case. First, since kH$ k kHk, the set H Ï H$ is nonempty and so we may adjoin a new element by setting H$ "ß! œ H$ ∪ Ö/$ × where /$ is the least element (under the well-ordering) not in H$ . Although H$"ß! may not have an upper bound, we can include upper bounds (under the partial order) for all finite subsets of H$"ß! . But this process must be repeated, since the inclusion of these upper bounds requires that additional upper bounds be included as well. Specifically, for 5 − , let H$ "ß5" œ H$ "ß5 ∪ Öan upper bound ?J for each finite subset J © H$ "ß5 × and let H$ " œ . H$ "ß5 5− Now, H$ " is directed since any Bß C − H$ " are also in some set H$ "ß5 and so have an upper bound in H$ "ß5" . Also, kH$ "ß5 k œ kH$ "ß! k œ kH$ k and so kH$ " k Ÿ i! † kH$ k œ kH$ k Ÿ k$ k œ k$ "k Moreover, (2.18) holds since if % $ " then % $ or % œ $ . If % $ , the induction hypothesis implies that .% − H$ © H$ " . If % œ $ , then since H$ contains all .% with % $ , it follows that if .$  H$ , then /$ œ .$ and so .$ − H$ "ß! © H$ " . Thus, the sequence ØHα ± α DÙ has the desired properties. Theorem 2.19 A poset T is complete if and only if it is chain-complete. 46 Lattices and Ordered Sets Proof. If T is complete, then it is chain-complete, since every nonempty chain is a directed set and since a complete poset has a smallest element by definition. For the converse, we use transfinite induction on the cardinality D of the directed set H. If D is finite, then H has a join, so we may assume that D œ i# is infinite and that all finite directed sets and all directed sets of cardinality iα for α # have a join. Theorem 2.17 implies that there is a strictly increasing transfinite sequence f œ ØHα ± α DÙ of directed subsets with kHα k kHk and .Hα œ H The induction hypothesis implies that each set Hα has a join +α . Since α " implies that Hα § H" , the sequence Ø+α ± α DÙ is a chain and so it has a join +. Since Hα Ÿ +α Ÿ +, it is clear that + is an upper bound for H. However, if H Ÿ , , then Hα Ÿ , and so +α Ÿ , , whence + Ÿ , . Thus, + is the join of H. Cofinality The following concept will prove useful in our study of the completeness (or rather the incompleteness) of lattices. Definition Let T be a poset. 1) A set E © T is cofinal in T if for any B − T , there is an + − E such that B Ÿ +, that is, if T œ ÆE A transfinite sequence Ø+α ± α " Ù is cofinal in T if the underlying set Ö+α ± α " × is cofinal in T . 2) Dually, a set E © T is coinitial if for any B − T there is an + − E such that + Ÿ B, that is, if T œ ÅE A transfinite sequence Ø+α ± α " Ù is coinitial in T if the underlying set Ö+α ± α " × is coinitial in T . If E is cofinal in T , then any upper bound ? for E must be in E and must be the largest element of T . In particular, if T has no largest element, then a cofinal set in T is unbounded. Well-Ordered Sets 47 Of course, a poset ÐT ß Ÿ Ñ is cofinal in itself. However, we are more interested in cofinal subsets that have some additional order properties. In particular, given a well-ordering of T via an injective transfinite sequence c œ Ø+α ± α $ Ù we can always find a subsequence V of c with the following properties: 1) The underlying set G of V is cofinal in T under the partial order, that is, for any : − T , there is an +α − V for which : Ÿ +α . 2) The well-order given by V is compatible with the partial order on T , that is, +α ß + " − V and α" Ê +α + " or +α ² +" In words, if +α comes before +" in the sequence V, then +α is less than or parallel to +" in the original partial order. The subsequence V is constructed by selecting the elements of c as follows: +" − V Í +" is Ÿ -maximal in Ö+α ± α Ÿ " × To see that the order of V is compatible with the partial order on T , if +α ß +" − V where α " , then the inclusion of +" implies that +" is maximal in a set that includes +α and so +α +" or +α ² +" . To see that G is cofinal in c , if +"  G , then +" is not maximal in Ø+α ± α Ÿ " Ù, which therefore contains an element greater than +" . But then the first such element (in sequential order) +$ is Ÿ -maximal in Ø+α ± α Ÿ $ Ù, since otherwise it would not be the first element greater than +" . Hence, +$ − G and +$ +" . Theorem 2.20 Let ÐT ß Ÿ Ñ be a poset, well-ordered by an injective transfinite sequence c œ Ø+α ± α $ Ù Then c has a cofinal subsequence V with the following properties: 1) The well-order of V is compatible with the order in ÐT ß Ÿ Ñ, that is, +α ß + " − V and α" Ê +α + " or +α ² +" 2) The underlying set G of V is cofinal in ÐT ß Ÿ Ñ, that is, if : − T then there is an +α − G for which : Ÿ +α . Exercises 1. 2. Give an example of a linearly ordered set T for which there is a proper down-set that is not an initial segment of T . Let [ be a well-ordered set for which ∞  [ . Let [ w œ [ ∪ Ö∞× and extend the order on [ so that + ∞ for all + − [ . Show that [ w is a well-ordered set. 48 3. 4. 5. 6. 7. 8. 9. 10. Lattices and Ordered Sets Find an example of a strictly monotone function that is not inflationary. Prove that a set \ is transitive if and only if -\ © \ . Prove that the intersection of transitive sets is transitive. Prove that the union of transitive sets is transitive. Prove that α " covers α for any α − ord. If α − ord show that bÐαÑ œ α ". Prove that if α − " − ord then either α " − " or α " œ " . Prove that the following are equivalent for an ordinal α: a) α is a limit ordinal b) " − ord, " − α Ê " " − α c) α is the union of all ordinals contained in α, that is, α œ ." " α 11. Prove that if a set W of ordinals has no greatest element, then -W is a limit ordinal. 12. Let 0 À [" ¸ [# be an isomorphism between well-ordered sets. a) Show that the restriction of 0 to [" Ò+Ó is an isomorphism 0 À [" Ò+Ó ¸ [# Ò0 Ð+ÑÓ. b) Show that the induced map 0 À kÐ[" Ñ Ä kÐ[# Ñ gives rise to a bijection between the families of initial segments, given by 0 Ð[" Ò+ÓÑ œ [# Ò0 Ð+ÑÓ for all + − [" . 13. Let Ð[ ß Ÿ Ñ be a well-ordered set and let ∞  [ . Let [ w œ [ ∪ Ö∞× and order [ w by extending the order on [ and specifying that ∞ + for all + − [ . Show that the order type of [ w is greater than that of [ . 14. Prove that any natural number is an initial ordinal. 15. Prove that an infinite cardinal number " is a limit ordinal. 16. Prove that if - is a limit ordinal then α - is also a limit ordinal. 17. Prove that the union of limit ordinals is a limit ordinal. 18. Prove that any ordinal α has the form α œ - 8 where - is a limit ordinal and 8 is a finite ordinal. Chapter 3 Lattices Closure and Inheritance As we have remarked, a nonempty subset W of a poset T inherits the order relation from T . However, W need not inherit existing meets and joins from T , in particular, a subset E © W may have a meet or join as a subset of W that is different from its meet or join as a subset of T . Closure We will generally avoid the somewhat ambiguous term closure and instead use the following terminology. Definition Let T be a poset. 1) T has finite meets if any nonempty finite subset of T has a meet. 2) T has arbitrary meets if any subset of T has a meet. 3) T has arbitrary nonempty meets if any nonempty subset of T has a meet. Dual definitions can be made with the word join in place of meet. If every pair of elements of T has a meet, then T has finite meets. A similar statement holds for joins. Inheritance As mentioned, if T is a poset, then a subset E © W © T may have a meet as a subset of W and a different meet as a subset of T . We refer to the meet of E as a subset of W as the W -meet of E and write 3W E, and similarly for T . Similar statements hold for joins as well. Definition A nonempty subset W of a poset T inherits finite meets from T if whenever the T -meet of a finite nonempty subset E © W exists, this T -meet is also in W (and is therefore the W -meet of E). Similar definitions can be made for the inheritance of arbitrary and arbitrary nonempty meets, as well as finite, arbitrary and arbitrary nonempty joins. S. Roman (ed.), Lattices and Ordered Sets, doi: 10.1007/978-0-387-78901-9_3, © Steven Roman 2008 49 50 Lattices and Ordered Sets Note that it is possible that a subset E of W has an W -meet but no T -meet. For example, consider the poset T œ ∪ Ö+ß ,×, where has the usual order and the incomparable elements + and , are greater than all members of . Then Ö+ß ,× has no meet in T but has a meet in W œ Ö!ß +ß ,×. Primeness A concept related to inheritance is primeness. Definition Let T be a poset and let W be a nonempty subset of T . 1) W is join-prime if +”, −W Ê +−W or , − W Ê +−W or , − W 2) Dually, W is meet-prime if +•, −W At the element level, we have the following definition. Definition 1) Let T be a poset with finite joins. An element B − T is join-prime if BŸ+”, Ê BŸ+ or B Ÿ , for all +ß , − T . 2) Dually, let T be a poset with finite meets. An element B − T is meet-prime if +•, ŸB Ê +ŸB or , Ÿ B for all +ß , − T . There is a simple connection between the two types of primeness. Theorem 3.1 Let T be a poset. 1) An element + − T is join-prime if and only if the principal filter Å B is joinprime. 2) Dually, an element + − T is meet-prime if and only if the principal ideal Æ B is meet-prime. Complementary Properties The connection between inheritance and primeness is also a simple one: In a word, they are complementary. Let us say that two properties : and ; of subsets of a poset T are complementary if W has property : Í T Ï W has property ; Theorem 3.2 Let T be a poset and let W © T . The following properties are complementary: Lattices 51 1) Up-set and down-set. 2) Inheriting finite meets and being meet-prime. 3) Inheriting finite joins and being join-prime. Semilattices Semilattices (and lattices) can be defined in two ways: one based on the existence of an order relation satisfying certain properties and one based on the existence of binary operations satisfying certain algebraic properties. The order-based definition of semilattice is the following. Definition 1) A poset T that has finite meets is called a meet-semilattice. 2) A poset T that has finite joins is called a join-semilattice. Note that a meet-semilattice cannot have two distinct minimal elements, since their meet contradicts the minimality of the elements. Hence, a meet-semilattice either has no minimal elements or else it has a unique minimal element, that is, a smallest element. In particular, a meet-semilattice with the DCC, such as a finite meet-semilattice, has a smallest element. Similar statements can be made for join-semilattices. Semilattices Defined Algebraically Semilattices can also be defined algebraically as follows. Definition A semilattice is a nonempty set T with a binary operation ‰ that satisfies the following properties: 1) (associativity) Ð+ ‰ ,Ñ ‰ - œ + ‰ Ð, ‰ -Ñ 2) (commutativity) +‰, œ,‰+ 3) (idempotency) +‰+œ+ Here is the connection between order semilattices and algebraic semilattices. Theorem 3.3 1) A meet-semilattice T is a semilattice under the meet operation. 2) A join-semilattice T is a semilattice under the join operation. 3) a) A semilattice T is a meet-semilattice with order defined by +Ÿ, if + ‰ , œ + 52 Lattices and Ordered Sets In this case, meet is +•, œ+‰, b) A semilattice T is a join-semilattice with order defined by +Ÿ, if + ‰ , œ , In this case, join is +”, œ+‰, Proof. Part 1) and part 2) are left to the reader. For part 3a), let T be a meetsemilattice. The idempotency law shows that + Ÿ + . If + Ÿ , and , Ÿ +, then +œ+‰, œ, Finally, if + Ÿ , and , Ÿ - , then + ‰ - œ Ð+ ‰ ,Ñ ‰ - œ + ‰ Ð, ‰ -Ñ œ + ‰ , œ + and so + Ÿ - . Thus, T is a poset. As to the meet, + ‰ , is a lower bound for + and , since Ð+ ‰ ,Ñ ‰ + œ Ð, ‰ +Ñ ‰ + œ , ‰ Ð+ ‰ +Ñ œ , ‰ + œ + ‰ , and so + ‰ , Ÿ +, and Ð+ ‰ ,Ñ ‰ , œ + ‰ Ð, ‰ ,Ñ œ + ‰ , and so + ‰ , Ÿ , . Also, if ? Ÿ + and ? Ÿ , , then ?‰+œ?œ?‰, and so ? ‰ Ð+ ‰ ,Ñ œ Ð? ‰ +Ñ ‰ , œ ? ‰ , œ ? whence ? Ÿ + ‰ , . Thus, + ‰ , is the greatest lower bound of + and , . The proof of part 3b) is similar. Arbitrary Meets Equivalent to Arbitrary Joins It may seem at first a bit surprising that a poset has arbitrary meets if and only if it has arbitrary joins. However, there is a simple explanation for this: The join of a family must be the meet of all elements that contain the members of the family (and dually). Theorem 3.4 A poset T has arbitrary meets if and only if it has arbitrary joins. In particular: 1) If T has arbitrary meets, then the join is given by Lattices 53 2E œ 4E ? for any subset E © T . 2) Dually, if T has arbitrary joins, then the meet is given by 4E œ 2E j for any subset E © T . Proof. Recall that for subsets \ and ] , the expression \ Ÿ ] means that B Ÿ C for all B − \ and C − ] . For part 1), let E be a nonempty subset of T . To see that 3E? is the least upper bound of E, we first note that E Ÿ E? Ê E Ÿ 4E ? so 3E? is an upper bound for E. Second, if B E, then B − E? , and so B 3E? , whence 3E? is the least upper bound of E, that is, the join of E. The previous theorem does not hold for closure under finite meets (or joins): If T is bounded and has finite meets, it does not necessarily have finite joins. A simple example is T œ Ö!ß +ß ,× where + ² , and ! is the smallest element of T . Here is a more interesting example, where T has a largest element. Example 3.5 Let T œ Ö!ß Bß Cß Ð"ß #Ó © ‘× be the poset with Ÿ defined as follows: 1) ! is the smallest element 2) B and C are less than any element of the half-open interval Ð"ß #Ó but not comparable to each other 3) the order in Ð"ß #Ó is the usual one for real numbers. Let W be a finite subset of T . If at least one of B or C is not in W , then the meet of W is the smallest element of W . If Bß C − W , then the meet of W is !. On the other hand, # is the largest element of T , but the set ÖBß C× has no join. Lattices We now come to the principal object of study of the rest of this book. Definition Let T be a poset. 1) T is a lattice if every pair of elements of T has a meet and a join. 2) T is a complete lattice if T has arbitrary meets and arbitrary joins. Note that a complete lattice is bounded. Also, we have seen that a poset T is complete if and only if it has arbitrary meets or arbitrary joins, in which case it has both. We remind the reader that the adjective complete has a different meaning when applied to posets than when applied to lattices. (Recall that a poset T is 54 Lattices and Ordered Sets complete or a CPO if T has a smallest element and if every directed subset of T , or equivalently by Theorem 2.19, every chain of T has a join.) Example 3.6 1) A singleton set P œ Ö+× is a lattice under the only possible order on P. This is a trivial lattice. Any lattice with more than one element is a nontrivial lattice. 2) Any totally ordered set is a lattice, but not necessarily a complete lattice. For example, the set ™ of integers under the natural order is a lattice, but not a complete lattice. 3) The set of natural numbers is a lattice under division, where + • , œ gcdÐ+ß ,Ñ and + ” , œ lcmÐ+ß ,Ñ 4) If W is a nonempty set, then the power set kÐWÑ is a complete lattice under union and intersection. 5) If T is a poset, then the collection bÐT Ñ of down-sets of T is a complete lattice, where meet is intersection and join is union. 6) If K is a group, then the family f ÐKÑ of subgroups of K is a complete lattice under set inclusion, where meet is intersection. Similar statements can be made for other algebraic objects, such as the ideals (or subrings) of a ring, the submodules of a module or the subfields of a field. 7) Let W be a nonempty set. The finite-cofinite algebra J GÐWÑ is the set of all subsets of W that are either finite or cofinite. (A subset E is cofinite if its complement is finite.) The set J GÐWÑ is a lattice under set inclusion and as we will see in a later chapter, it is also a Boolean algebra. If a lattice P does not have a !, then we can adjoin a ! to the poset P. The set P¼ is also a lattice since finite meets and finite joins of elements of P are not affected and since ! • + œ ! and ! ” + œ +. Of course, the adjoining of a smallest element does create additional arbitrary meets. Similarly, we can adjoin a " if P does not have ". The Direct Product of Lattices The cartesian product of any nonempty family of lattices is a lattice under componentwise operations. Definition Let Y œ ÖP3 ± 3 − M× be a nonempty family of lattices. The direct product #P3 is the lattice consisting of the cartesian product of the family Y , with componentwise operations, that is, if 0 ß 1 − #P3 , then Ð0 • 1ÑÐ3Ñ œ 0 Ð3Ñ • 1Ð3Ñ and Ð0 ” 1ÑÐ3Ñ œ 0 Ð3Ñ ” 1Ð3Ñ Lattices 55 Meet-Structures and Closure Operators Before discussing the properties of lattices, let us take a look at one important source of these structures. Definition 1) Let \ be a nonempty set. A subset f of the power set kÐ\Ñ is called an intersection-structure on \ , abbreviated ∩ -structure, if f inherits arbitrary intersections from kÐ\Ñ, that is, if ÖW3 ± 3 − M× © f Ê , ÖW3 × © f 3−M 2) More generally, a subset W of a complete lattice P is called a meetstructure, abbreviated • -structure, if W inherits arbitrary meets from P. Note that some authors define an ∩ -structure as a subset that inherits arbitrary nonempty intersections. Intersection-structures are also called closure systems, but we prefer to avoid this rather nonspecific terminology. Since a meet-structure has arbitrary meets, it is a complete lattice. The main reason why meet-structures are important is that they characterize the closed sets of a closure operator, and closure operators are quite common in mathematics. Definition A closure operator on a poset T is a unary operator cl on T with the following properties: 1) (extensive) : Ÿ clÐ:Ñ 2) (monotone) :Ÿ; Ê clÐ:Ñ Ÿ clÐ;Ñ 3) (idempotent) clÐclÐ:ÑÑ œ clÐ:Ñ The element clÐ:Ñ is called the closure of :. An element : − T is closed if it is equal to its own closure, that is, if : œ clÐ:Ñ. We denote the closed elements in T by ClÐT Ñ. A closure operator on a power set kÐ\Ñ is often referred to as a closure operator on \ . We leave it to the reader to check that clÐ:Ñ is the smallest closed element of T containing : and that if " − T , then " is closed. 56 Lattices and Ordered Sets Now let P be a complete lattice. To see that ClÐT Ñ is a meet-structure, it is clear that the empty meet (being the largest element) is closed and if Y œ Ö53 ± 3 − M× is a nonempty family in ClÐPÑ, then 4 53 Ÿ 5 4 P for all 4. Taking closures gives clŠ4P 53 ‹ Ÿ clÐ54 Ñ œ 54 and so 4 54 Ÿ clŠ4 53 ‹ Ÿ 4 54 P P P which shows that 3P 53 is closed. Thus, ClÐPÑ is a meet-structure. It follows that the closure of an element + − P is equal to the meet of all closed elements containing +, that is, clÐ+Ñ œ 4 Ö5 − ClÐPÑ ± + Ÿ 5× Thus, the family ClÐPÑ of closed sets completely determines the closure operator, that is, the map >À cl È ClÐPÑ is an injection from closure operators on P to meet-structures in P. To see that > is surjective, we take the hint from the previous display. Theorem 3.7 If G is a meet-structure in a complete lattice P, the G -closure of + − P, defined by + œ 4 Ö- − G ± + Ÿ -× is a closure operator on P. Proof. It is easy to see that this map is extensive and monotone. As to idempotence, we have + Ÿ Ð+Ñ œ 4Ö- − G ± + Ÿ -× Ÿ + where the final inequality holds because + is a term in the meet. Thus, the map >À cl È ClÐPÑ is a bijection from closure operators on P to meetstructures in P and its inverse is given by the G -closure. To see this, if cl is a closure operator on P and + È + is the ClÐPÑ-closure operator corresponding to Lattices ClÐPÑ, then 57 + œ 4 Ö- − ClÐPÑ ± + Ÿ -× œ clÐ+Ñ Theorem 3.8 Let P be a complete lattice. 1) The maps >À cl Ä ClÐPÑ and CÀ G Ä G -closure are inverse bijections between the closure operators on P and the meetstructures on P. 2) If cl is a closure operator on P, then the ClÐPÑ-join of E © ClÐPÑ is the closure of the P-join of E, that is, 2 ClÐPÑ E œ clŠ2P E‹ Proof. For part 2), the right hand side is an upper bound for E in P that happens to be in ClÐPÑ and so 2 ClÐPÑ E Ÿ clŠ2P E‹ Ÿ clŠ2ClÐPÑ E‹ œ 2ClÐPÑ E Example 3.9 Let K be a group and let T be the collection of nonempty subsets of K. Set clÐWÑ œ ØWÙ, the subgroup generated by W . Then cl is a closure operator on T , with corresponding ∩ -structure f ÐKÑ, the complete lattice of subgroups of K. Put another way, the closed subsets of K are the subgroups of K. Note that if L and O are subgroups of K, then the union L ∪ O is not closed in general. /7 Example 3.10 Let 8 œ :"/" â:7 be the prime decomposition of a positive integer. Let P œ Ð8 ß ± Ñ be the lattice of all positive integers less than or equal to 8, under division. Fix an integer " Ÿ 5 Ÿ 7. For + − P, let cl5 Ð+Ñ be the integer formed by replacing the exponent of :5 in + by /5 . Then cl5 is a closure operator on P. In loose terms, cl5 is “max out the exponent of :5 .” Thus, the closed integers + are those for which :5/5 ± +. Note that if + and , are closed, then + • , œ gcdÐ+ß ,Ñ and + ” , œ lcmÐ+ß ,Ñ are also closed. Example 3.11 Let Ð\ß 7 Ñ be a topological space. The topological closure operator E È E is a closure operator in the sense defined above and so the closed subsets of \ form an ∩ -structure, as is well known. Closure operators on a power set kÐ\Ñ can be divided into two classes. Definition Let \ be a nonempty set. A closure operator cl on kÐ\Ñ is finitary if for any W © \ , 58 Lattices and Ordered Sets clÐWÑ œ .ÖclÐJ Ñ ± J © Wß J finite× A closure operator that is not finitary is infinitary. The closure operators generally encountered in algebra are finitary. For example, the subgroup generated by a nonempty subset W of a group K is the union of the finitely-generated (even cyclic) subgroups generated by elements of W . However, if Ð\ß 7 Ñ is a X" topological space (points are closed), then any finite set is closed and so the closure operator is finitary if and only if for any W © \, W œ .ÖJ ± J © Wß J finite× œ W Of course, this holds if and only if all sets are closed, that is, if and only if 7 is the discrete topology. In summary, for a X" space, topological closure is finitary if and only if the topology is discrete. We will have much more to say about the types of closure operators, and corresponding meet-structures, related to the substructures of an algebraic structure in a later chapter. Galois Connections We next describe an important construction that leads to closure operators. Definition Let T and U be posets. A Galois connection on the pair ÐT ß UÑ is a pair ÐCß HÑ of maps CÀ T Ä U and HÀ U Ä T , where we write CÐ:Ñ œ : ‡ and HÐ;Ñ œ ; w , with the following properties: 1) (Order-reversing) For all :ß ; − T and <ß = − U, : Ÿ ; Ê :‡ ; ‡ and < Ÿ = Ê < w =w and ; Ÿ ; w‡ 2) (Extensive) For all : − T , ; − U, : Ÿ :‡w Example 3.12 If T is a poset, then the maps ?ß jÀ kÐT Ñ Ä kÐT Ñ defined by ?ÐEÑ œ E? and jÐEÑ œ Ej for any E © T form a Galois connection on the power set kÐT Ñ, that is, the pair Ð?ß jÑ is a Galois connection for ÐkÐT Ñß kÐT ÑÑ. Example 3.13 Let \ and ] be nonempty sets and let V © \ ‚ ] be a relation on \ ‚ ] . Then the maps W © \ È W ‡ œ ÖC − ] ± BVC for all B − W× and Lattices 59 X © ] È X w œ ÖB − \ ± BVC for all C − X × form a Galois connection on ÐkÐ\Ñß kÐ] ÑÑ. Example 3.14 Let J and I be fields with J © I . Let K œ KÐIÎJ Ñ be the group of all automorphisms of I that fix each element of J . The group K is called the Galois group of the field extension J Ÿ I . The most famous example of a Galois connection is the Galois correspondence between the intermediate fields J © O © I and the subgroups L of the Galois group K. This correspondence is given by O È KÐIÎOÑ ³ Ö5 − K ± 5B œ B for all B − O× and L È fixÐLÑ ³ ÖB − I ± 5B œ B for all 5 − L× Example 3.15 Let 8 " and let J be a field. Let J ÒB" ß á ß B8 Ó denote the ring of polynomials in 8 variables over J and let J 8 denote the ring of all 8-tuples over J . Let T œ kÐJ ÒB" ß á ß B8 ÓÑ and U œ kÐJ 8 Ñ Let CÀ J ÒB" ß á ß B8 Ó Ä c ÐJ 8 Ñ be defined by CÐWÑ œ Set of all common roots of the polynomials in W œ ÖB − J 8 ± :ÐBÑ œ ! for all : − W× and let HÀ c ÐJ 8 Ñ Ä J ÒB" ß á ß B8 Ó be defined by HÐX Ñ œ Set of all polynomials whose root set includes X œ Ö: − J ÒB" ß á ß B8 Ó ± :Ð>Ñ œ ! for all > − X × The pair ÐCß HÑ is a Galois connection on ÐT ß UÑ. There are two closure operators associated with a Galois connection. Theorem 3.16 Let ÐCß HÑ be a Galois connection on ÐT ß UÑ. Then : ‡w ‡ œ : ‡ and ; w‡w œ ; w for all : − T and ; − U. It follows that the composite maps : Ä :‡w and ; Ä ; w‡ are closure operators on T and U, respectively. Furthermore, 1) :‡ is closed for all : − T . 2) ; w is closed for all ; − U. 3) The maps CÀ T Ä ClÐUÑ and HÀ U Ä ClÐT Ñ are surjective and the restricted maps CÀ ClÐT Ñ Ä ClÐUÑ and HÀ ClÐUÑ Ä ClÐT Ñ are inverse 60 Lattices and Ordered Sets bijections. Thus, ClÐT Ñ œ Ö; w ± ; − U× and ClÐUÑ œ Ö:‡ ± : − T × Proof. Since : Ÿ :‡w , the order-reversing property of * gives :‡w‡ Ÿ :‡ Ÿ Ð:‡ Ñw‡ and so :‡w‡ œ :‡ . We leave the rest of the proof to the reader. Properties of Lattices We now resume our discussion of the properties of lattices. Lattices as Algebraic Structures Since a lattice P is a meet-semilattice and a join-semilattice, Theorem 3.3 implies that meet and join are associative, commutative and idempotent operations. The link between the two operations is provided by the absorption laws + • Ð+ ” ,Ñ œ + and + ” Ð+ • ,Ñ œ + In fact, these properties characterize lattices. Theorem 3.17 The following properties hold in a lattice P: L1) (associativity) Ð+ ” ,Ñ ” - œ + ” Ð, ” -Ñ Ð+ • ,Ñ • - œ + • Ð, • -Ñ L2) (commutativity) +”, œ,”+ +•, œ,•+ L3) (idempotency) +”+œ+ +•+œ+ L4) (absorption) + ” Ð+ • ,Ñ œ + + • Ð+ ” ,Ñ œ + Conversely, if P is a nonempty set with two binary operations • and ” satisfying L1)–L4), then P is a lattice where meet is • , join is ” and the order relation is given by Lattices +Ÿ, if + ” , œ , +Ÿ, if + • , œ + 61 or equivalently, Moreover, since the set of axioms L1)–L4) is self-dual, it follows that if a statement holds in every lattice, then any dual statement holds in every lattice. Proof. As to the equivalence of the statements + ” , œ , and + • , œ +, the absorption laws imply that +•, œ+ Ê + ” , œ Ð+ • ,Ñ ” , œ , +”, œ, Ê + • , œ + • Ð+ ” ,Ñ œ + and Next we show that Ÿ is a partial order. The idempotency law implies that + Ÿ +. If + Ÿ , and , Ÿ +, then , œ+”, œ+ For transitivity, if + Ÿ , and , Ÿ - , then + ” - œ + ” Ð, ” -Ñ œ Ð+ ” ,Ñ ” - œ , ” - œ and so + Ÿ - . Finally, Theorem 3.3 implies that • and ” are the meet and join in P. Varieties of Lattices Theorem 3.17 shows that lattices can be defined equationally, that is, by identities. More generally, a variety or equational class of lattices is a family of lattices defined by a set of lattice identities. Theorem 3.17 says that the family of all lattices is an )-based variety of lattices, since there are ) identities in Theorem 3.17. As another example, distributive lattices are those lattices P that satisfy the two distributive identities + • Ð, ” -Ñ œ Ð+ • ,Ñ ” Ð+ • -Ñ + ” Ð, • -Ñ œ Ð+ ” ,Ñ • Ð+ ” -Ñ for all +ß ,ß - − P. Thus, the distributive lattices form a "!-based variety of lattices. There has been work done on identifying minimal sets of identities for varieties of lattices. Padmanabhan [49] has shown that if i is a variety of lattices defined by a finite number of identities, then i can also be defined by just two identities. Also, McKenzie [45] has shown that lattice theory itself, that is, the family of all lattices, is actually " -based. In other words, there is a single 62 Lattices and Ordered Sets identity that characterizes lattices among all equationally-based “algebras” with two binary operations. However, as McKenzie writes “… although the resulting equation to characterize lattices (equation - below) is so long, containing over three hundred thousand symbols, that we can only represent it in abbreviated form. This leaves open the problem of finding a really elegant equation to characterize lattices.” McKenzie has also shown that no other variety of lattices, except the variety at the other extreme, in which all equations are identities, is "-based. Tops and Unbounded Sequences If a lattice P has no largest element, then P must be infinite and it seems reasonable that P should have some form of strictly increasing infinite sequence with no upper bound. This is easy to see when P is countably infinite. For if P œ Ð+" ß +# ß á Ñ has no largest element, the sequence of finite joins f œ Ø=5 Ù œ Ø+" ” â ” +5 Ù has no upper bound, since f Ÿ ? implies that P Ÿ ?. Removing all duplicates from f gives a strictly increasing infinite sequence with no upper bound. For an arbitrary infinite lattice P with no largest element, we can well-order P and then Theorem 2.20 implies that P contains a cofinal transfinite subsequence f œ Ø,α ± α $ Ù compatible with the order of P, that is, ,α , " Ê ,α Ÿ , " or ,α ² ," Let us assume that all duplicates have been removed from this subsequence and so ,α , " Ê ,α , " or ,α ² ," To eliminate the problem of parallel elements, let ^ be a maximal chain in f (with respect to the order of P), which must exist by the Hausdorff maximality principle. Then ^ also has no upper bound in P. For if ^ Ÿ ? for some ? − P, then ? must be the largest element of P. Otherwise, there is an + − P for which ? + or ? ² +. In either case, ^ Ÿ ? ? ” + and the cofinality of f in P implies that there is an = − f for which ^ ? ” + Ÿ =. Hence, ^ ∪ Ö=× is a chain in f , which contradicts the maximality of ^. Thus, ^ is a strictly increasing transfinite sequence in P with no upper bound. Theorem 3.18 1) A countably infinite lattice P has no largest element if and only if P has a nonempty strictly increasing sequence with no upper bound in P. 2) An infinite lattice P has no largest element if and only if it has a nonempty strictly increasing transfinite sequence Lattices 63 U œ Ø,α ± α ,Ù for some ordinal ,, with no upper bound in P. Of course, the previous theorem has a dual, which we leave to the reader to formulate. Bounds for Subsets On a related matter, given a nonempty subset of a lattice P, can we find a strictly increasing sequence of elements of P that has the same set of upper bounds as E? This is trivial if E is a finite set: Just take the one-element sequence consisting of the join 1E. For a countably infinite set E, we can sequentially order E to get E œ Ð+" ß +# ß á Ñ, then take the sequence of finite joins as before f œ Ø=5 Ù œ Ø+" ” â ” +5 Ù Removing all duplicate entries gives a sequence that has the same upper bounds as E. If E is an infinite set of arbitrary cardinality kEk œ D, then we can well-order E to get a transfinite sequence T œ Ø+α ± α DÙ of length D. Since D is an initial ordinal, if % D, then k%k kDk. Thus, to mimic the proof for the countably infinite case, we assume that every nonempty subset of E of cardinality less than D has a join. Then for any % D, the transfinite subsequence T% œ Ø+α ± α %Ù of T has a join ?% œ 2Ø+α ± α %Ù Moreover, the transfinite sequence of joins U œ Ø?% ± % DÙ is nondecreasing and U ? œ E? . As before, we can cast out any duplicate entries in U to get a strictly increasing sequence. Theorem 3.19 Let P be a lattice. 1) If E œ Ö+" ß +# ß á × 64 Lattices and Ordered Sets is a countably infinite subset of P, then there is a strictly increasing sequence f in P for which f ? œ E? . 2) If E is an infinite subset of P of cardinality kEk œ D and if every nonempty subset of E of cardinality less than D has a join in P, then there is a strictly increasing transfinite sequence f œ Ø=α ± α " Ù in P for which f ? œ E? . Join-Irreducible and Meet-Irreducible Elements The following types of elements play a special role in the representation theory of lattices. Definition Let P be a lattice. 1) A nonzero element 4 − P is join-irreducible if 4 is not the join of two smaller elements, that is, if 4œ+”, Ê 4œ+ or 4 œ , The set of join-irreducible elements of P is denoted by ] ÐPÑ. 2) Dually, a nonunit element 7 − P is meet-irreducible if 7 is not the meet of two larger elements, that is, if 7œ+•, Ê 7œ+ or 7 œ , The set of meet-irreducible elements of P is denoted by `ÐPÑ. Example 3.20 1) The atoms of any lattice are join-irreducible. 2) In a chain, all nonzero elements are join-irreducible. 3) In the power set lattice kÐ\Ñ, the join-irreducible elements are the singleton subsets. 4) In Ðß ± Ñ an element is join-irreducible if and only if it is a prime power. However, Ðß ± Ñ has no meet-irreducible elements, since for example 8 œ #8 • $8. 5) In the lattice of all open subsets of the plane (under set inclusion), no elements are join-irreducible. 6) If T is a finite poset, the join-irreducible elements of bÐT Ñ are precisely the principal ideals in bÐT Ñ. It is clear that Æ : is join-irreducible, since if Æ : œ H" ∪ H# , then : − H3 for some 3 and so Æ : œ H3 . Also, if H − bÐT Ñ is join-irreducible, then since H œ . Ö Æ .× .−H is a finite join, it follows that H œ Æ . for some . − H and so H is principal. Lattices 65 Completeness Let us take a closer look at completeness. Chain Conditions and Completeness Chain conditions are a form of finiteness condition. One aspect of that finiteness is that the chain conditions imply that an arbitrary join (or meet) is equal to a finite join (or meet), as described in the following theorem. Theorem 3.21 Let P be a lattice. 1) If P has the ACC, then P has arbitrary nonempty joins. In fact, for every nonempty subset E © P there is a finite subset E! © E such that 2E œ 2E! Thus, a lattice with ! and with ACC is complete. 2) Dually, if P has the DCC then P has arbitrary nonempty meets. In fact, for every nonempty subset E © P there is a finite subset E! © E such that 4E œ 4E! Thus, a lattice with " and with DCC is complete. Proof. For part 1), suppose that P has the ACC and let E be a nonempty subset of P. Consider the family Y of all joins of finite subsets of E. Since Y is nonempty, it has a maximal element, say 7 œ 2 E! Now if 1E! is not an upper bound for E, there is an + − E for which 1E! + or 1E! ² +. In either case, 2ÐE! ∪ Ö+×Ñ 2E! which contradicts the maximality of 1E! . Hence, 1E! is an upper bound for E. Since it is clearly a least upper bound for E, we have 1E œ 1E! . Note that the converse does not hold: A complete lattice need not have the ACC. Indeed, the bounded lattice ™‡ consisting of the integers under the usual order with a smallest and a largest element adjoined is complete but has neither chain condition. Incompleteness and Coalescing Pairs of Sequences It is not hard to find conditions that characterize the incompleteness of an infinite lattice P. Again, we consider the countable case first, since this does not involve the use of ordinals or transfinite sequences. 66 Lattices and Ordered Sets The Countable Case Let P be a countably infinite lattice. Of course, if P is missing either a largest element or a smallest element, then P is not complete. We have seen that the lack of a largest element is equivalent to the existence of a strictly increasing infinite sequence in P with no upper bound. A similar statement holds for smallest elements. Intuitively speaking, the presence of a strictly increasing sequence with no upper bound indicates that P is missing a “limit point” for the sequence. But a lattice can be missing other limit points as well. This happens when there is a strictly increasing sequence and a strictly decreasing sequence that are headed toward each other on a collision course and, while they never actually meet, they leave no space in between. This happens, for example, in the lattice of all nonzero rational numbers (under the usual order) for the sequences Ð"Î8Ñ and Ð"Î8Ñ. For convenience, we make the following nonstandard definition. Definition Let P be a lattice. A strictly increasing (possibly empty) sequence F œ Ð," ,# âÑ and a strictly decreasing (possibly empty) sequence G œ Ðâ -# -" Ñ coalesce if the following hold: 1) F G , that is, ,3 -4 for all 3ß 4. 2) There is no + − P for which F Ÿ + Ÿ G , or equivalently, F? ∩ G j œ g In this case, we say that ÐFß GÑ is a coalescing pair in P. Note that if ÐFß GÑ is a coalescing pair, then F cannot have a join and G cannot have a meet. In particular, F and G are either infinite or else empty. Moreover, F is empty if and only if G is a strictly decreasing sequence with no lower bound, that is, if and only if P has no smallest element. Dually, G is empty if and only if P has no largest element. Now we can characterize incompleteness in a countably infinite lattice. Theorem 3.22 A countably infinite lattice P is incomplete if and only if it has a coalescing pair ÐFß GÑ of sequences. Proof. If P is bounded but incomplete, then there is an infinite subset W œ Ö=" ß =# ß á × Lattices 67 of P that has no join in P. Since P has a largest element, the set W ? is nonempty and is, in fact, inherits meets and joins from P and so is a lattice. Moreover, W ? has no smallest element, since such an element would be the join of W in P. Now, we may apply Theorem 3.19 to W to get a strictly increasing sequence F œ Ð," ,# âÑ for which F ? œ W ? . Then we may apply the dual of Theorem 3.18 to F ? to get a strictly decreasing infinite sequence G œ Ð-" -# âÑ with no lower bound in F ? . Thus, F ? ∩ G j œ g. It is also clear that F G and so the pair ÐFß GÑ is a coalescing pair. The General Case The general case is entirely analogous. Definition Let P be a lattice. A strictly increasing (possibly empty) transfinite sequence U œ Ø,α ± α ,Ù and a strictly decreasing (possibly empty) transfinite sequence V œ Ø-α ± α $ Ù coalesce if the following hold: 1) U V, that is, ,α -" for all α , and " $ . 2) There is no + − P for which U Ÿ + Ÿ V, or equivalently, U ? ∩ Vj œ g We refer to the pair ÐU ß VÑ as a coalescing pair in P. As in the countable case, if ÐU ß VÑ is a coalescing pair, then U cannot have a join and V cannot have a meet. Hence, U and V are either infinite or else empty. Moreover, U is empty if and only if P has no smallest element and V is empty if and only if P has no largest element. Theorem 3.23 (Davis, 1955 [13]) A lattice P is incomplete if and only if P has a coalescing pair ÐU ß VÑ. Proof. If P is bounded but not complete, then there is a nonempty subset of P that has no join. Pick such a subset of least cardinality, say W œ Ö=3 ± 3 − M× Thus, any subset of W of smaller cardinality has a join and Theorem 3.19 implies that there is a strictly increasing transfinite sequence 68 Lattices and Ordered Sets U œ Ø,α ± α $ Ù in W that has no join in P. Since P is bounded, the set U ? is nonempty and inherits meets and joins from P and so is a lattice. Moreover, since U has no join, F ? has no smallest element. Hence, Theorem 3.18 implies that U ? has a strictly decreasing transfinite sequence V œ Ø-α ± α ,Ù with no lower bound in U ? and so U ? ∩ Vj œ g. It is clear that U V and so ÐU ß VÑ is a coalescing pair. Chain-Completeness A lattice P is chain-complete if every chain in P, including the empty chain, has a join. The previous characterization of incompleteness makes it easy to show that a lattice is complete if and only if it is chain-complete. Theorem 3.24 A lattice is complete if and only if it is chain-complete. Proof. Assume that P is chain-complete but not complete. Since P is chaincomplete, it has a smallest element. Hence, Theorem 3.23 implies that P has a coalescing pair ÐU ß VÑ, where U is nonempty. However, since U is a chain in P, it has a join, which is not possible. Hence, P must be complete. Sublattices A nonempty subset Q of a lattice P inherits the order of P, but not necessarily the meets and joins. When it does, it is called a sublattice of P. Definition Let P be a lattice and let Q © P be a nonempty subset of P. 1) Q is a sublattice of P if Q inherits finite meets and finite joins from P. 2) If P is complete, then Q is a complete sublattice of P if Q inherits arbitrary meets and arbitrary joins from P. Example 3.25 Let P œ Ö"ß #ß $ß 'ß "#× under division (see Figure 3.1). 12 6 3 2 1 Figure 3.1 Lattices 69 The subset W œ Ö"ß #ß $ß "#× is a lattice under division but not a sublattice of P. The subset X œ Ö"ß #ß $ß '× is a sublattice of P. Example 3.26 If P is a lattice and + , , then the closed interval Ò+ß ,Ó is a sublattice of P. Definition Let W be a nonempty set. A sublattice of a power set kÐWÑ is called a power set sublattice, or a ring of sets. Thus, in a power set sublattice, meet is intersection and join is union. For example, the down-sets bÐT Ñ of a poset form a power set sublattice. On the other hand, the family fÐZ Ñ of subspaces of a vector space Z is a lattice, where meet is intersection but join is in general not union. Hence, fÐZ Ñ is not a power set sublattice. If P is a lattice, then the intersection of any family of sublattices of P is either empty or it is a sublattice. Thus, for any nonempty subset E © P, there is a smallest sublattice of P containing E: It is just the intersection of all sublattices containing E. Definition If P is a lattice and E © P is nonempty, then the sublattice generated by E, denoted by ÒEÓ, is the smallest sublattice of P containing E. Lattice Terms We can also describe the sublattice generated by a nonempty subset in more constructive terms using lattice polynomials and evaluation maps. Intuitively speaking, ÒEÓ is simply the set of all “lattice legal” expressions that can be formed using the elements of E and the meet and join operators (and parentheses). But we wish to be a bit more formal. Definition Let \ be a nonempty set. A lattice term (or just term), also called a lattice polynomial, over \ is any expression defined as follows. Let “ and ’ be formal symbols. 1) The elements of \ are terms of weight ". 2) If : and ; are terms, then so are Ð: ’ ;Ñ and Ð: “ ;Ñ whose weights are the sum of the weights of : and ; . 3) An expression in the symbols \ ∪ Ö ’ ß “ ß Ðß Ñ× is a term if it can be formed by a finite number of applications of 1) and 2). We denote the set of all lattice terms over \ by g\ . The elements of \ are called variables. Two terms are equal if and only if they are identical. Thus, for example B" “ B# is not equal to B# “ B" . 70 Lattices and Ordered Sets It is customary to omit the final pair of parentheses when writing lattice terms. A lattice term involving some or all of the variables B3" ß á ß B38 − \ but no others is denoted by :ÐB3" ß á ß B38 Ñ. We now define two binary operations • and ” , called meet and join respectively, on g\ . For :ß ; − g\ , let :•; œ:’; and : ” ; œ : “ ; Then since B3 ’ B4 œ B3 • B4 for all B3 ß B4 − \ and similarly for join, we can write any lattice term : − g\ using only the variables and the meet and join symbols (and parentheses). Note that we do not claim that g\ is a lattice under meet and join. Indeed, g\ consists of formal expressions and so, for example, B" ” B# Á B# ” B" . Now, given a lattice P and a function 0 À \ Ä E, where E © P, the evaluation map %0 À g\ Ä P is defined, for each :ÐB3" ß á ß B38 Ñ − g\ by %0 :ÐB3" ß á ß B38 Ñ œ :Ð0 B3" ß á ß 0 B38 Ñ Note that for lattice terms : and ; , %0 Ð: • ;Ñ œ %0 : • %0 ; and %0 Ð: ” ;Ñ œ %0 : ” %0 ; We can now describe the sublattice ÒEÓ as the set of all elements of P that are images of all possible lattice terms under all possible evaluation maps into E. Theorem 3.27 Let P be a lattice and let E © P be nonempty. Let \ œ ÖB" ß B# ß á ×. Then ÒEÓ œ Ö%0 :ÐB" ß á ß B8 Ñ ± : − g\ ß 0 À \ Ä E× Proof. Let W be the set on the right above. It is clear that any element of W is in ÒEÓ and since W is a sublattice of P, it follows that W œ ÒEÓ. Denseness Definition Let H be a nonempty subset of a poset T . 1) H is join-dense in T if for every : − T , there is a subset W © H for which : œ 2T W 2) H is finite-join-dense in T if for every : − T , there is a finite nonempty subset W © H for which : œ 2T W Meet-dense and finite-meet-dense are defined dually. Lattices 71 Note that if H is join-dense in T , then any : − T is the join of all elements of H that are contained in :, that is, : œ 2HÆ Ð:Ñ Example 3.28 It is clear that the singleton sets are join-dense in the power set kÐ\Ñ. However, if \ is infinite, then the singleton sets are not finite-joindense. In a lattice with the DCC, the join-irreducible elements are finite-join-dense. Theorem 3.29 Let P be a lattice. 1) If P has the DCC, then ] ÐPÑ is finite-join-dense in P. 2) Dually, if P has the ACC, then `ÐPÑ is finite-meet-dense in P. Proof. If some element of P is not the join of a finite number of join-irreducible elements, then let + be a minimal such element. Thus, + is not join-irreducible and so it has the form + œ , ” - where , + and - +. The minimality of + implies that , and - are joins of a finite number of join-irreducible elements and therefore so is +, which is a contradiction. Lattice Homomorphisms A monotone map 0 À P Ä Q between lattices need not, in general, preserve meets and joins. Even an order embedding need not preserve meets and joins. Example 3.30 Consider the lattices of integers in Figure 3.2, where the order is division. 12 12 6 3 2 3 2 1 1 L M Figure 3.2 The inclusion map 0 À P Ä Q defined by 0 Ð8Ñ œ 8 is an order embedding, but 0 Ð# ” $Ñ œ 0 Ð"#Ñ œ "# and 0 Ð#Ñ ” 0 Ð$Ñ œ # ” $ œ ' Hence, 0 does not preserve joins. 72 Lattices and Ordered Sets A monotone map 0 À P Ä Q does send a lower bound for a subset W © P to a lower bound of 0 ÐWÑ and so 0 Š4W ‹ Ÿ 40 ÐWÑ and similarly 0 Š2W ‹ 20 ÐWÑ assuming that the relevant meets and joins exist. In fact, each of these conditions is equivalent to 0 being monotone. For example, if + Ÿ , , then + • , œ + and so 0 Ð+Ñ œ 0 Ð+ • ,Ñ Ÿ 0 Ð+Ñ • 0 Ð,Ñ Ÿ 0 Ð,Ñ On the other hand, it is easy to see that an order isomorphism does preserve meets and joins. Definition Let P and Q be lattices. A function 0 À P Ä Q that preserves finite meets and joins, that is, for which 0 Ð+ • ,Ñ œ 0 Ð+Ñ • 0 Ð,Ñ 0 Ð+ ” ,Ñ œ 0 Ð+Ñ ” 0 Ð,Ñ is called a lattice homomorphism. 1) A lattice monomorphism or lattice embedding is an injective lattice homomorphism. 2) A lattice epimorphism is a surjective lattice homomorphism. 3) A lattice endomorphism of P is a lattice homomorphism from P to itself. 4) A lattice isomorphism is a bijective lattice homomorphism, or equivalently, an order isomorphism. A lattice homomorphism 0 À P Ä Q of bounded lattices for which 0 Ð!Ñ œ ! and 0 Ð"Ñ œ " is called a Ö!ß "×-homomorphism. To reiterate, we have the following (and no more): lattice embedding lattice isomorphism Ê Í order embedding order isomorphism A key property of lattice embeddings not shared by order embeddings is the following. Theorem 3.31 Let 0 À P Ä Q be a lattice embedding. 1) 0 ÐPÑ is a sublattice of Q . 2) If P is complete, then so is 0 ÐPÑ, but 0 ÐPÑ need not be a complete sublattice of Q . Proof. For part 2), the map 0 À P Ä 0 ÐPÑ is a lattice isomorphism. We leave it to the reader to show that 0 ÐPÑ is also complete. On the other hand, let Lattices 73 P œ ∪ Ö∞× have the usual order on and let 8 ∞ for all 8 − . Let Q œ ∪ Ö+ß ∞× with the usual order on and 8 + ∞ for all 8 − . Let 0 À P Ä Q be the inclusion map, which is a lattice embedding. However, 0 Š2P ‹ œ 0 Ð∞Ñ œ ∞ but 2 0 Ð Ñ œ 2 œ + Á ∞ Q Q Meet and Join Homomorphisms Definition Let P and Q be lattices. 1) A function 0 À P Ä Q that preserves finite meets, that is, 0 Ð+ • ,Ñ œ 0 Ð+Ñ • 0 Ð,Ñ is called a meet-homomorphism. 2) Dually, a function 0 À P Ä Q that preserves finite joins, that is, 0 Ð+ ” ,Ñ œ 0 Ð+Ñ ” 0 Ð,Ñ is called a join-homomorphism. A meet-homomorphism is order preserving, since +Ÿ, Í +•, œ+ Ê 0 Ð+Ñ • 0 Ð,Ñ œ 0 Ð+Ñ Í 0 Ð+Ñ Ÿ 0 Ð,Ñ Dually, a join-homomorphism is order preserving as well. The F -Down Map The F -down map 9FßÆ À P Ä kÐFÑ, defined by 9FßÆ Ð+Ñ œ FÆ Ð+Ñ ³ Ö, − F ± , Ÿ +× is a meet-homomorphism, since FÆ Ð+ • ,Ñ œ FÆ Ð+Ñ ∩ FÆ Ð,Ñ but it is not a join-homomorphism, since in general FÆ Ð+ ” ,Ñ Á FÆ Ð+Ñ ∪ FÆ Ð,Ñ Of course, the F -down map is order preserving, that is, BŸC Ê FÆ ÐBÑ © FÆ ÐCÑ but it need not be an order embedding. Theorem 3.32 Let F be a nonempty subset of a lattice P and let 9FßÆ À P Ä kÐFÑ be the F -down map. The following are equivalent: 74 Lattices and Ordered Sets 1) 9FßÆ is an order embedding, that is, for all Bß C − P, BŸC Í FÆ ÐBÑ © FÆ ÐCÑ 2) 9FßÆ satisfies the following for all Bß C − P, BC Í FÆ ÐBÑ § FÆ ÐCÑ 3) 9FßÆ is injective, equivalently, for all Bß C − P, BœC Í FÆ ÐBÑ œ FÆ ÐCÑ 4) 9FßÆ preserves strict order, that is, for all Bß C − P, BC Ê FÆ ÐBÑ § FÆ ÐCÑ 5) F is join-dense in P. If these conditions hold, then F is said to be a base for P. Proof. We first show that 1) and 5) are equivalent. If 5) holds and FÆ ÐBÑ © FÆ ÐCÑ, then B œ 2FÆ ÐBÑ Ÿ 2FÆ ÐCÑ œ C and since the F -down map is always order-preserving, it follows that 1) holds. Conversely, if 1) holds, then FÆ ÐBÑ Ÿ B and if FÆ ÐBÑ Ÿ ?, then FÆ ÐBÑ © FÆ Ð?Ñ and so 1) implies that B Ÿ ?. Thus, B œ 1FÆ ÐBÑ. To complete the proof, we show that 2) Ê 4) Ê 1) Ê 3) Ê 2). Note first that FÆ ÐBÑ © FÆ ÐCÑ Í FÆ ÐBÑ œ FÆ ÐB • CÑ (3.33) It is clear that 2) implies 4). If 4) holds and FÆ ÐBÑ © FÆ ÐCÑ, then (3.33) implies that FÆ ÐBÑ œ FÆ ÐB • CÑ. Hence, 4) shows that B • C B cannot hold and so B • C œ B, that is, B Ÿ C. Thus, 4) implies 1). It is easy to see that 1) implies 3). Finally, (3.33) shows that 3) implies 1) and together they imply 2). One often sees the definition of a base F given in the form BŸ y C Í b, − F with , Ÿ B and , Ÿ y C for all Bß C − P. Of course, this is equivalent to BŸC Í FÆ ÐBÑ © FÆ ÐCÑ Ideals and Filters The following are important substructures in lattices. Definition Let P be a lattice. 1) An ideal of P is a nonempty down-set in P that inherits finite joins, that is, a) + − Mß B Ÿ + Ê B − M Lattices 75 b) +ß , − M Ê + ” , − M In this case, we write M ü P. A proper ideal, that is, an ideal M Á P, is denoted by M – P. The set of all ideals of P is denoted by \ ÐPÑ. 2) Dually, a filter in P is a nonempty up-set in P that inherits finite meets, that is, a) + − J ß B + Ê B − J b) +ß , − J Ê + • , − J In this case, we write J ý L. A proper filter, that is, a filter J Á P, is denoted by J — P. The set of all filters of P is denoted by Y ÐPÑ. Note that the properties of being an ideal and being a filter are not complementary. Example 3.34 If P is a lattice and + − P, then the principal ideal Æ + œ ÖB − P ± B Ÿ +× is an ideal of P and the principal filter Å + is a filter. The intersection M ∩ N of two ideals M and N of a lattice P is not empty, since if 3 − M and 4 − N , then 3 • 4 − M ∩ N . Hence, M ∩ N is an ideal of P. On the other hand, in the lattice ™ of all integers under the usual order, the intersection of the family Ö Æ 8 ± 8 − ™× of ideals is empty. This leaves us with two alternatives. We can deal only with lattices with ! for which \ ÐPÑ is an ∩ -structure or we can define \g ÐPÑ œ \ ÐPÑ ∪ Ög× which is an intersection-structure for any lattice P. Theorem 3.35 Let P be a lattice. 1) The set \ ÐPÑ is a lattice, where meet is intersection and join is given by M ” N œ ÖB − P ± B Ÿ 3 ” 4 for some 3 − M and 4 − N × 2) The set \g ÐPÑ is an ∩ -structure and therefore a complete lattice, where the join of a family Y of ideals is given by 2Y œ ÖB − P ± B Ÿ +" ” â ” +8 , where +3 − .Y and 8 "× 3) The down map 9Æ À P Ä \g ÐPÑ is a lattice embedding, that is, for all +ß , − P, Æ + ∩ Æ , œ Æ Ð+ • ,Ñ and Æ + ” Æ , œ Æ Ð+ ” ,Ñ The image of 9Æ is the sublattice c ÐPÑ of \g ÐPÑ consisting of all principal ideals of P. 4) The union of any directed family of ideals in P is an ideal in P. 76 Lattices and Ordered Sets 5) If P has the ACC, then all ideals are principal. Dually, if P has the DCC then all filters are principal. Proof. It is clear that the set M œ ÖB − P ± B Ÿ +" ” â ” +8 , where +3 − .Y × is an ideal containing M5 and so 1Y © M . Also, since +" ” â ” +8 − 1Y , it follows that M © 1Y and so M œ 1Y . For part 3), if it is easy to see that Æ + ∩ Æ , œ Æ Ð+ • ,Ñ and Æ + ” Æ , œ ÖB − P ± B Ÿ 3 ” 4 for 3 − Æ + and 4 − Æ ,× œ ÖB − P ± B Ÿ + ” ,× œ Æ Ð+ ” ,Ñ For part 4), let V be a directed family of ideals in \ ÐPÑ and let Y œ -V . If + − Y and , Ÿ +, then + − G − V and so , − G © Y . Also, if +ß , − Y , then + and , both lie in one of the ideals G − V and so + ” , − G © Y . For part 5), Theorem 3.21 implies that if M is an ideal, then there is a finite subset M! of M for which 1M! œ 1M . Hence, M œ Æ 1M! is principal. The Ideal Generated by a Subset Every nonempty subset E of a lattice P is contained in a smallest ideal, namely, the intersection of all ideals of P containing E. Definition Let E be a nonempty subset of a lattice P. 1) The ideal generated by E, denoted by ÐEÓ, is the smallest ideal of P containing E. 2) The filter generated by E, denoted by ÒEÑ, is the smallest filter of P containing E. We leave proof of the following to the reader. Theorem 3.36 Let P be a lattice and let E © P be a nonempty subset of P. 1) ÐEÓ œ Æ ÒEÓ œ Æ gE 2) ÐEÓ œ 2Ö Æ + ± + − E× œ ÖB − P ± B Ÿ +" ” â ” +8 for +3 − E× Lattices 77 Prime and Maximal Ideals The following special types of ideals play a key role in lattice theory. Definition Let P be a lattice. 1) A proper ideal M of P is maximal if for any ideal M , N ©M©P Ê MœN or M œ P A proper filter J of P is maximal if for any filter \ , J ©\©P Ê \œJ or \ œ P A maximal filter is also called an ultrafilter. 2) A proper ideal N is prime if +•, −N Ê +−N or , − N The set of prime ideals of P is called the spectrum of P and is denoted by SpecÐPÑ. Dually, a proper filter J is prime if +”, −J Ê +−J or , − J Although the properties of being an ideal and being a filter are not complementary, Theorem 3.2 does imply the following. Theorem 3.37 The properties of being a prime ideal and being a prime filter are complementary, that is, M is a prime ideal if and only if M - is a prime filter. We leave proof of the following to the reader. Theorem 3.38 Let 0 À P Ä Q be a lattice homomorphism. 1) If M ü Q , then 0 " ÐMÑ ü P. 2) If T − SpecÐQ Ñ, then 0 " ÐT Ñ − SpecÐPÑ. In ring theory, maximal ideals are prime. This is not true in general for lattices. Example 3.39 In the lattice Q$ of Figure 3.3, 1 x y 0 M3 Figure 3.3 z 78 Lattices and Ordered Sets the ideal M œ Ö!ß B× is maximal. However, C • D − M but C  M and D  M and so M is not prime. Ideal Kernels and Homomorphisms It is usual in discussions of homomorphisms of the common algebraic structures, such as groups, rings and modules, to discuss the kernel of a homomorphism. We will postpone a careful discussion of kernels of lattice homomorphisms to a later chapter, but make a few remarks here in connection with prime ideals. If a lattice Q has a smallest element, then the ideal kernel of a lattice homomorphism 0 À P Ä Q is the set kerÐ0 Ñ œ 0 " Ð!Ñ œ Ö+ − P ± 0 Ð+Ñ œ !× This is easily seen to be an ideal of P. There is another type of kernel, related to congruence relations, that will be discussed in a later chapter. This accounts for the adjective ideal in the term ideal kernel. On the other hand, not all ideals are ideal kernels. Example 3.40 Consider again the lattice Q$ in Figure 3.3, of which we will have much to say throughout the book. The subset M œ Ö!ß D× is an ideal of Q$ . Suppose that 0 À Q$ Ä Q is a lattice homomorphism for which 0 Ð!Ñ œ 0 ÐDÑ œ !. Then 0 Ð"Ñ œ 0 ÐC ” DÑ œ 0 ÐCÑ ” 0 ÐDÑ œ 0 ÐCÑ Hence, ! œ 0 Ð!Ñ œ 0 ÐB • CÑ œ 0 ÐBÑ • 0 ÐCÑ œ 0 ÐBÑ • 0 Ð"Ñ œ 0 ÐB • "Ñ œ 0 ÐBÑ and so B − kerÐ0 Ñ. This shows that M is not the kernel of any lattice homomorphism. We will discuss in more detail the question of when an ideal of a lattice is a kernel of some homomorphism in a later chapter. Nevertheless, we can address the issue now for prime ideals. If P is a lattice and W © P, then the indicator function of W is the function 0W À P Ä Ö!ß "× defined by setting 0W ÐBÑ œ ! if and only if B − W . Here we consider the set Ö!ß "× as a two-element lattice with ! ". In general, an indicator function need not be a lattice homomorphism. However, the indicator function of a prime ideal T is a lattice epimorphism, since Lattices 79 0T Ð+ • ,Ñ œ ! Í + • , − T Í + − T or , − T Í 0T Ð+Ñ œ ! or 0T Ð,Ñ œ ! Í 0T Ð+Ñ • 0T Ð,Ñ œ ! and since T - is a prime filter, 0T Ð+ ” ,Ñ œ " Í + ” , − T Í + − T - or , − T Í 0T Ð+Ñ œ " or 0T Ð,Ñ œ " Í 0T Ð+Ñ ” 0T Ð,Ñ œ " Of course, the ideal kernel of 0T is T . Conversely, suppose that the indicator function 0W À P Ä Ö!ß "× is an epimorphism. Then W œ kerÐ0W Ñ œ 0 " Ö!× is a prime ideal in P, since Ö!× is a prime ideal in Ö!ß "×. Thus, prime ideals can be characterized as the ideal kernels of indicator epimorphisms. Theorem 3.41 In a lattice P, the prime ideals in P are precisely the ideal kernels of the indicator epimorphisms. Lattice Representations A lattice homomorphism 0 À P Ä Q is often referred to as a representation map, or simply as the representation of P in Q . The most useful representations are the embeddings, since in this case, P is isomorphic to a sublattice of Q . Embeddings are also called faithful representations. The advantages of a faithful representation appear when Q provides a more familiar setting than P. For example, intersections and unions of sets are more familiar than general meets and joins. Thus, it is often desirable to represent the elements of a lattice P by subsets of a set \ , with meet and join represented by intersection and union, if possible. When representing elements of a lattice by subsets of a set \ , the set \ is called the carrier set for the representation. As we will see, various versions of the F -down map 9FßÆ À + È Æ +, obtained by varying the set F and the range of the map, are valuable tools for finding lattice representations. For instance, Theorem 3.35 implies the following. Theorem 3.42 (Representation theorem for lattices) For any lattice P, the down map 9Æ À P ä \g ÐPÑ 80 Lattices and Ordered Sets is a lattice embedding. If P is complete, then 9Æ ÐPÑ is an intersection-structure and so every complete lattice is isomorphic to an intersection-structure. Proof. The second statement follows from Theorem 3.31. Note that in the range \g ÐPÑ of this version of the down map, meet is intersection but join is not in general union. We can change the range of the map so that meet is intersection and join is union by replacing \g ÐPÑ with bÐPÑ: 9Æ À P Ä bÐPÑ However, this map is a meet-homomorphism but not, in general, a joinhomomorphism. Indeed, an arbitrary lattice cannot be faithfully represented as a power set sublattice, since power set sublattices are distributive (defined below). On the other hand, we will see in a later chapter that an arbitrary distributive lattice can be faithfully represented as a power set sublattice, via yet another variation of the down map. Special Types of Lattices There are many special types of lattices. We introduce the most important types here and go into more details in later chapters. Atomic and Atomistic Lattices Recall that an atom in a lattice P with ! is a element + − P that covers !. Definition 1) A lattice P is atomic if every nonzero element of P contains an atom. 2) A lattice P is atomistic if every nonzero element of P is the join of atoms. It is clear that atomistic lattices are atomic. However, a finite chain of length at least $ is atomic but not atomistic. For the record, we note some of the more common variations on the concept of atomicity: Definition 1) A lattice P is strongly atomic if every nontrivial interval Ò+ß ,Ó in P is atomic, that is, if for any + , in P, there exist ? − P for which +ö?Ÿ, 2) A lattice P is weakly atomic if for any + , in P, there exist ?ß @ − P for which +Ÿ?ö@Ÿ, Lattices 81 Distributive Lattices It is not hard to show that if a lattice P satisfies either one of the following distributive laws: + • Ð, ” -Ñ œ Ð+ • ,Ñ ” Ð+ • -Ñ + ” Ð, • -Ñ œ Ð+ ” ,Ñ • Ð+ ” -Ñ for all elements in P, then it satisfies the other distributive law. We also note that for each distributive law, one side is greater than or equal to the other in all lattices. For instance, we always have + • Ð, ” -Ñ Ð+ • ,Ñ ” Ð+ • -Ñ since each term on the left is greater than or equal to each term on the right. Definition A lattice P for which the distributive laws hold is called a distributive lattice. It is well known that any ring of sets is a distributive lattice, since E ∩ ÐF ∪ GÑ œ ÐE ∩ FÑ ∪ ÐE ∩ GÑ If we adjoin a new largest element or smallest element to a distributive lattice P, the resulting lattice is still distributive. Also, we leave it as an exercise to show that if P is distributive, then so is the lattice \ ÐPÑ. As we will see, the distributive property is quite a strong property, making distributive lattices much better behaved than their nondistributive cousins. One example is that in a distributive lattice, all maximal ideals are prime. In fact, in some sense, distributive lattices are closer in behavior to Boolean algebras than to nondistributive lattices. In particular, in 1933, Garrett Birkhoff [6] published his famous representation theorem for distributive lattices, which says that a lattice is distributive if and only if it is isomorphic to a power set sublattice (ring of sets). Thus, for distributive lattices, meet is essentially intersection and join is essentially union. Modular Lattices The modular property is a special case of the distributive property. Definition A lattice P is modular if it satisfies the modular law: For all +ß ,ß - − P, + - Ê + • Ð, ” -Ñ œ Ð+ • ,Ñ ” - Note that any distributive lattice is modular. Also, we leave it as an exercise to show that if P is modular, then so is the lattice \ ÐPÑ. 82 Lattices and Ordered Sets One reason for the importance of modular lattices is that some very important lattices are modular but not distributive. For example, the lattice of normal subgroups a ÐKÑ of a group K is modular. (This will be easy to prove in the next chapter.) On the other hand, it is not hard to see that the lattice of (normal) subgroups of the Klein four-group is isomorphic to Q$ , which is not distributive. We also note here that the lattice fÐKÑ of all subgroups of a group need not be modular; the alternating group E% is an example. Note that the modular law is equivalent to the identity + • Ð, ” Ð+ • -ÑÑ œ Ð+ • ,Ñ ” Ð+ • -Ñ for all +ß ,ß - − P. This identity is obtained from the consequent of the modular law by replacing - by + • . , which represents an arbitrary element contained in + (and then replacing . by - ). Thus, modularity, like distributivity defined below, is a variety of lattices. Complemented Lattices Lattice elements may have complements. Definition Let P be a bounded lattice. A complement of + − P is an element , − P for which +•, œ! and + ” , œ " In this case, we say that + and , are complementary. Complements need not exist and they need not be unique when they do exist. For instance, in a bounded chain, no elements other than ! and " have complements. Also, in both of the lattices in Figure 3.4, the element C has two complements. 1 x 1 y x z y z 0 M5 0 N5 Figure 3.4 However, we will see that if a lattice is distributive, then complements, when they exist, are unique. Definition Lattices 83 1) A bounded lattice P for which each element has a complement is called a complemented lattice. 2) A bounded lattice P for which each element has a unique complement is called a uniquely complemented lattice. It is also easy to see that in a complemented lattice any prime ideal is maximal. In particular, in a complemented distributive lattice, otherwise known as a Boolean algebra, the prime ideals are the same as the maximal ideals. Here are a few simple properties of complements that hold in uniquely complemented lattices that do not hold in arbitrary complemented lattices. Theorem 3.43 Let P be a uniquely complemented lattice. 1) If + − P then +ww œ +. 2) If + is an atom in P, then +w is a coatom. 3) If + and , are distinct atoms in P, then + Ÿ , w . These statements do not hold in arbitrary complemented lattices. Proof. The lattices Q$ and R& show that these statements do not hold in arbitrary complemented lattices (where +w is interpreted as any complement of +). Part 1) follows from the definition of complement. For part 2), if +w œ ", then + œ +ww œ "w œ !, which is false. Also, if +w Ÿ B ", then + ” B + ” +w œ " and so + ” B œ ". Also, since + is an atom, + • B œ ! or + • B œ +. But + • B œ + implies that + Ÿ B, which gives " œ + ” +w Ÿ B which is false. Hence, + • B œ ! and so B œ +w . Thus, " covers +w , that is, +w is a coatom. For part 3), since , w is a coatom, + ” , w œ , w or + ” , w œ ". In the former case, + Ÿ , w , as desired. In the case + ” , w œ ", we cannot have + • , w œ !, since then +w œ , w , which implies that + œ , . Thus, + • , w œ +, that is, + Ÿ , w . Relatively Complemented Lattices Relative complements are a generalization of complements. Definition Let ? @ be elements of a lattice P and let + − Ò?ß @Ó. A relative complement of + with respect to Ò?ß @Ó is a complement of + in the sublattice Ò?ß @Ó, that is, an element B − Ò?ß @Ó for which B•+œ? and B ” + œ @ The set of all relative complements of + with respect to Ò?ß @Ó is denoted by +Ò?ß@Ó . If the relative complement is unique, we will simply refer to it as +Ò?ß@Ó . Use of the notation +Ò?ß@Ó tacitly implies that + − Ò?ß @Ó. Theorem 3.44 Let P be a lattice. The following are equivalent: 84 Lattices and Ordered Sets 1) Relative complements, when they exist, are unique. 2) An element of P is uniquely determined by its meet and join with any other element, that is, for all Bß +ß , − P B•+œB•, œB ” + œ B ” , Ê +œ, Definition 1) A lattice P is relatively complemented if every closed interval Ò?ß @Ó in P is complemented. 2) A lattice P is uniquely relatively complemented if every closed interval Ò?ß @Ó in P is uniquely complemented. A complemented lattice need not be relatively complemented. For instance, in the lattice R& of Figure 3.4, the element D has no relative complement with respect to the interval Ò!ß BÓ. On the other hand, this cannot happen in a modular lattice. Theorem 3.45 Let P be a bounded modular lattice. If B − P has a complement Bw , then B also has a relative complement B< with respect to any interval Ò+ß ,Ó containing B. In fact, B< is formed by projecting Bw into the interval Ò+ß ,Ó as follows: B< œ ÐBw ” +Ñ • , œ ÐBw • ,Ñ ” + Proof. Since , +, the modular law shows that the two expressions above are equal. Also, B • ÒÐBw • ,Ñ ” +Ó œ ÒB • ÐBw • ,ÑÓ ” + œ + and B ” ÒÐBw • ,Ñ ” +Ó œ B ” ÐBw • ,Ñ œ ÐB ” Bw Ñ • , œ , and so ÐBw • ,Ñ ” + − BÒ+ß,Ó . Sectionally Complemented Lattices In some cases, all we need in order to prove a result is that the lattice in question has relative complements with respect to intervals of the form Ò!ß @Ó. Definition A lattice P is sectionally complemented if it has a smallest element and if every interval of the form Ò!ß @Ó in P is complemented. Recall that any atom in a lattice is join-irreducible. In sectionally complemented lattices, the converse is also true. Also, a sectionally complemented lattice is atomistic if and only if it is atomic. Theorem 3.46 Let P be a sectionally complemented lattice. Lattices 85 1) An element + − P is join-irreducible if and only if it is an atom, that is, ] ÐPÑ œ TÐPÑ 2) P is atomistic if and only if it is atomic. Proof. For part 1), we have TÐPÑ © ] ÐPÑ in any lattice. If + is not an atom, then there is a , − P for which ! , +. If - − ,Ò!ß+Ó , then - ” , œ + and since - • , œ !, it follows that - Á +. Hence, + is not join-irreducible. Thus, ] ÐPÑ © TÐPÑ. For part 2), assume that P is atomic. For B Á !, it is clear that TÆ ÐBÑ Ÿ B. If TÆ ÐBÑ Ÿ ?, then TÆ ÐBÑ Ÿ 7 ³ B • ? Ÿ B and 7 !. But if ! 7 B, then 7 has a relative complement 7w in Ò!ß BÓ and ! 7w B. Since P is atomic, there is an atom + 7w B and so + − TÆ ÐBÑ. Hence, + Ÿ 7 and so + Ÿ 7 • 7w œ !, which is false. Thus, 7 œ B, which is equivalent to B Ÿ ?. This shows that B œ 1TÆ ÐBÑ and so P is atomistic. The Dedekind–MacNeille Completion Any poset T can be order embedded in a complete lattice P. When this happens, it is customary to say that P is a completion of T . For example, the down map 9Æ À T Ä bÐT Ñ is an order embedding of T into bÐT Ñ. However, although the image 9Æ ÐT Ñ, which is the family of principal ideals, is join-dense in bÐT Ñ, it is not generally meet-dense. On the other hand, by restricting the range bÐT Ñ of the down map, we can get an order embedding of T into a complete lattice in such a way that the image is both join-dense and meet-dense in the range. To this end, note that for any + − T, Æ + œ +?j where ?ß jÀ kÐT Ñ Ä kÐT Ñ are the maps defined by ?ÐEÑ œ E? and jÐEÑ œ Ej for any E © T . For readability, we also write Ö+×? as +? and similarly for j. Thus, the image of the down map belongs to the family of all subsets of T of the form E?j for E © T . Now, it is easy to see that the pair Ð?ß jÑ is a Galois connection on kÐT Ñ, that is, for the pair ÐkÐT Ñß kÐT ÑÑ. Therefore, the composition ?j is a closure operator on T and so the family of subsets of the form E?j is the family of closed subsets of this closure operator and is therefore an ∩ -structure. 86 Lattices and Ordered Sets Theorem 3.47 Let T be a poset. The maps ?ß jÀ kÐT Ñ Ä kÐT Ñ defined by ?ÐEÑ œ E? and jÐEÑ œ Ej for any E © T form a Galois connection on kÐT Ñ, that is, the following hold: 1) For any Eß F © T , E © Ej? and F © F ?j 2) For any Eß F © T , E©F Ê F j © Ej and F ? © E? Hence: 3) For all E © T , Ej?j œ Ej and E?j? œ E? 4) The compositions j? and ?j are closure operators on kÐT Ñ and so the families of closed sets G?j ³ ÖW © T ± W œ W ?j × and Gj? ³ ÖW © T ± W œ W j? × are ∩ -structures and hence complete lattices. The complete lattice of closed sets under the closure operator ?j has a name. Definition Let T be a poset. The ∩ -structure consisting of the closed sets of the operator ?j: DMÐT Ñ œ ÖE © T ± E?j œ E× œ ÖE?j ± E © T × is called the Dedekind–MacNeille (DM) completion, the MacNeille completion, the completion by cuts or the normal completion of T . The next theorem justifies the term completion for DMÐT Ñ. Theorem 3.48 Let T be a poset and let DMÐT Ñ be its Dedekind–MacNeille completion. Let 9Æ À T Ä DMÐT Ñ be the down map with range DMÐT Ñ, that is, 9Æ Ð+Ñ œ Æ + œ +?j for all + − T . 1) The image 9Æ ÐT Ñ is both join-dense and meet-dense in DMÐT Ñ. In particular, for any E © T , Lattices 87 E?j œ ,Ö9Æ ÐBÑ ± B − E? × and E?j œ 2DMÐT Ñ Ö9Æ Ð+Ñ ± + − E× 2) 9Æ preserves existing meets and joins in T , that is, if E © T has a meet, then 9Æ Š4T E‹ œ ,Ö9Æ Ð+Ñ ± + − E× and if E has a join, then 9Æ Š2T E‹ œ E?j œ 2DMÐT Ñ Ö9Æ Ð+Ñ ± + − E× In particular: a) If T is a lattice, then 9Æ À T ä DMÐT Ñ is a lattice embedding. b) If T is a complete lattice, then 9Æ À T ¸ DMÐT Ñ is an isomorphism. Proof. For the first equation in part 1), we have α − E?j Í α Ÿ E? Í α Ÿ B for all B − E? Í α − B?j for all B − E? Í α − ,ÖB?j ± B − E? × For the second equation, since + − E implies that +?j © E?j , it follows that E?j is an upper bound for the family Y ³ Ö+?j ± + − E× in DMÐT Ñ. But if F is an upper bound for Y in DMÐT Ñ, then E © F and so E?j © F ?j œ F . Hence, E?j is the least upper bound of Y . The first statement in 2) is clear and the second statement follows from part 1), using the fact that 9Æ Š2T E‹ œ Æ Š2T E‹ œ E?j Moreover, if T is complete, then for any E − DMÐT Ñ, we have E œ E?j œ 9Æ Š2T E‹ and so 9Æ is surjective. Theorem 3.48 says that T is order isomorphic to a join-dense and meet-dense subset 9Æ ÐT Ñ of DMÐT Ñ. Moreover, if P is a lattice, then 9Æ ÐPÑ is a sublattice of DMÐT Ñ and if P is complete, then 9Æ ÐPÑ œ DMÐPÑ. This is what we expect from a true completion of T . 88 Lattices and Ordered Sets The Dedekind–MacNeille completion DMÐT Ñ is characterized, up to isomorphism, by the fact that it is a complete lattice P in which T is both joindense and meet-dense. Theorem 3.49 Let P be a complete lattice. If a subset T © P is both join-dense and meet-dense in P, then P ¸ DMÐT Ñ, via the T -down map 9T ßÆ À P ¸ DMÐT Ñ defined by 9T ßÆ Ð+Ñ œ TÆ Ð+Ñ Proof. We must first show that TÆ Ð+Ñ − DMÐT Ñ, that is, that TÆ Ð+Ñ?T jT œ TÆ Ð+Ñ However, since T is join-dense in P, we have + œ 2TÆ Ð+Ñ and so B − TÆ Ð+Ñ?T Í B − T and B + Í B − TÅ Ð+Ñ that is, TÆ Ð+Ñ?T œ TÅ Ð+Ñ Similarly, the meet-denseness of T in P implies that TÆ Ð+Ñ?T jT œ TÅ Ð+ÑjT œ TÅ Ð+Ñj ∩ T œ ’4TÅ Ð+Ñ“ ∩ T j œ TÆ Ð+Ñ Thus, 9T ßÆ does map P into DMÐT Ñ. Moreover, 9T ßÆ is an order embedding, since +" Ÿ + # Í TÆ Ð+" Ñ © TÆ Ð+# Ñ Í 9T ßÆ Ð+" Ñ © 9T ßÆ Ð+# Ñ For surjectivity, the elements E − DMÐT Ñ satisfy E œ E?T jT , where E © T . But if + œ 1E, then E œ E?T jT œ +?T jT œ TÅ Ð+ÑjT œ TÆ Ð+Ñ œ 9T ßÆ Ð+Ñ and so 9T ßÆ is surjective. Example 3.50 Consider the rational numbers with the usual order, as a subset of the real numbers ‘. Adjoin a largest element ∞ and a smallest element ∞ to ‘ and let ‘w œ Ö∞ß ∞× ∪ ‘. Then ‘w is a complete lattice. Lattices Moreover, for any < − ‘, we have < œ 2‘ Æ Ð<Ñ and ∞ œ 2‘ and and 89 < œ 4‘ Å Ð<Ñ ∞ œ 4‘ and so is both join-dense and meet-dense in ‘w . Hence, ‘w is isomorphic to the Dedekind–MacNeille completion of . Let us remark that if P is a distributive lattice, it does not necessarily follow that its Dedekind–MacNeille completion is distributive. Funayama [21] (see also page 238 of Balbes [1]) has found a rather involved example of a distributive lattice P for which the completion DMÐPÑ has a sublattice isomorphic to the nondistributive lattice R& and is therefore not distributive. Exercises 1. 2. 3. 4. 5. Prove that if P is a distributive lattice, then any maximal ideal is prime. Prove that if P is a complemented lattice then any prime ideal of P is maximal. Let 0 À P Ä Q be a lattice embedding. Show that 0 ÐPÑ is a sublattice of Q . What are the largest and smallest elements of the lattice Ðß ± Ñ? If ÐT ß Ÿ Ñ is a meet-semilattice then ÐT ß ‰ Ñ is a semilattice where +‰, œ+•, 6. 7. 8. 9. What does the corresponding join-semilattice look like? Prove that the set ™‡ of integers, under the usual order and with a smallest element ∞ and a largest element ∞ adjoined, is complete but has neither chain condition. Show that the following are equivalent for a lattice homomorphism 0 À P Ä Q between uniquely complemented lattices: a) 0 preserves complements, that is, 0 Ð+w Ñ œ Ð0 +Ñw for all + − P. b) 0 is a Ö!ß "×-homomorphism. What are the join-irreducible elements of Й ß Ÿ Ñ? What are the joinirreducible elements of Й ß ± Ñ? Let E œ Ö+" ß +# ß á × and F œ Ö," ß ,# ß á × be disjoint countably infinite sets. Define a relation V on the union E ∪ F by +3" V+3 ß ,3" V,3 and ,3 V+3 for 3 œ "ß #ß á Þ Show that V is a covering relation. Draw a picture of the corresponding poset P. Show that P is a lattice. Find the join-irreducible elements. Find an element of P that is not the join of join-irreducibles. 10. Find a lattice P that has no join-irreducible elements, other than the one in the text. 90 Lattices and Ordered Sets 11. Let P be a lattice and let + − P. Let \ be a generating set for P. For any subset W of P, let T ÐWÑ be the property that for any finite subset W! © W , + Ÿ 2W! Ê + = for some = − W! Show that if T Ð\Ñ holds then T ÐPÑ holds. Hint: Show that if T ÐJ Ñ holds for some \ © J , then for any ?ß @ − J , it follows that T ÐJ ∪ Ö? ” @×Ñ and T ÐJ ∪ Ö? • @×Ñ hold. 12. Let T be a finite poset. Theorem 1.11 says that for each finite poset T , the down map 9T ßÆ À T Ä bÐT Ñ defined by 9T ßÆ Ð+Ñ œ Æ + is an order embedding whose image is the set of principal ideals in bÐT Ñ: imÐ9T ßÆ Ñ œ Ö Æ + ± + − T × Prove that the principal ideals are precisely the join-irreducibles in bÐT Ñ and so the map 9T ßÆ À T ¸ ] ÐbÐT ÑÑ is an order isomorphism. 13. Let ÐCß HÑ be a Galois connection on ÐT ß UÑ. If T and U are lattices, show that de Morgan's laws hold in ClÐT Ñ and ClÐUÑ, that is, for :ß ; − ClÐT Ñ and <ß = − ClÐUÑ, Ð: • ;ч œ :‡ ” ; ‡ ß Ð: ” ;ч œ :‡ • ; ‡ and Ð< • =Ñw œ <w ” =w ß Ð< ” =Ñw œ <w • =w 14. A lattice P is atomistic if every nonzero element of P is the join of atoms. What is the relationship, if any, between atomic and atomistic? 15. Let P be a lattice. Show that if + − P then the join map 4+ À P Ä P defined by 4+ B œ B ” + is monotone. Is it an order embedding? 16. Prove that if we adjoin a ! (or ") to a distributive lattice P that does not have a smallest element (or largest element), the resulting lattice is still distributive. Prove similar statement concerning modular lattices. 17. Let T be a poset and let Ö+3ß4 ± 3 − Mß 4 − N × be a doubly-indexed subset of T . Show that if for all ? − M and @ − N , the joins 2 +?ß4 , 2 +3ß@ 4 3 and exist then 13ß4 +3ß4 and 14 13 +3ß4 exist and 2 Š2 +3ß4 ‹ 3 4 2 +3ß4 œ 2 Š2 +3ß4 ‹ œ 2 Š2 +3ß4 ‹ 3ß4 3 4 4 3 18. Prove that the down map 9Æ À P Ä bÐPÑ is a lattice homomorphism. 19. Let P be a lattice. Prove that M © P is an ideal of P if and only if M is a sublattice and that 3 − M , + − P imply + • 3 − M . Lattices 91 20. Prove that a lattice P has the property that every ideal is prime if and only if P is a chain. 21. Let P be a distributive lattice. Prove that if + − P has a complement +w then for any prime ideal M , exactly one of + or +w is in M . 22. Let P be a lattice. a) Show that if P is modular, then so is \ ÐPÑ. b) Show that if P is distributive, then so is \ ÐPÑ. 23. If 0 À P Ä Q is a lattice homomorphism, show that 0 " ÐMÑ ü P for every M ü Q. 24. Prove that if 5À T Ä U is an order isomorphism, its induced map 5À bÐT Ñ Ä bÐUÑ is a lattice isomorphism. 25. Let M be an ideal of a lattice P and let +  M . Describe the smallest ideal containing M and +. 26. Let P be a lattice and let Eß F © P be nonempty. Prove that Æ ÒEÓ ∩ Å ÒFÓ œ g if and only if for all nonempty finite subsets E! © E and F! © F , we have 1E! y 3F! . 27. Let Z be a vector space and let 5À Z Ä Z be a linear operator on Z . Show that the family of 5 -invariant subspaces of Z is an ∩ -structure. 28. Describe the ideals of Й ß ± Ñ. 29. Prove that a nonempty subset W of a lattice P is a convex sublattice of P if and only if W œ M ∩ J , where M is an ideal and J is a filter. 30. Let P be a lattice. a) Let W be a nonempty subset of P. Show that W is contained in a directed subset H of P with the same join as W , that is, 1W exists if and only if 1H exists, in which case the two are equal. b) Use part a) to prove that if P is chain-complete then P is complete. 31. Let P be a lattice. Prove that the following are equivalent. a) Relative complements, when they exist, are unique. b) An element of P is uniquely determined by its meet and join with any another element. That is, for all +ß ,ß - − P +•, œ+•œ+ ” , œ + ” - Ê ,œ- 32. Let T be a poset and let bÐT Ñ be the poset of down-sets in T , under set inclusion. a) Show that any principal ideal Æ + is join-irreducible in bÐT Ñ. b) If T is finite, show that ] Ðb ÐT ÑÑ is the set of principal ideals. 33. Let P be a relatively complemented lattice. Prove the following: a) α − +Òαß" Ó Í + œ " b) " − +Òαß" Ó Í + œ α c) + − +Òαß" Ó Í + œ α œ " 34. Let P be a lattice. Prove that the congruences classes of a congruence relation on P are convex sublattices of P. 35. A lattice P with ! is disjunctive if + Ÿ y , implies that there is an B − P such that 92 Lattices and Ordered Sets B•+! and B • , œ ! Prove that a lattice P with ! has a base consisting of atoms of P if and only if P is atomic and disjunctive. 36. Prove directly (without using the concept of a base) that if P is a lattice with the DCC then ] ÐPÑ is join-dense in P. The Dedekind–MacNeille Completion 37. Let \ be a nonempty set. Find the smallest family Y of subsets of \ for which DMÐY Ñ ¸ kÐ\Ñ. 38. Find the Dedekind–MacNeille completion of an antichain T . 39. Let \ be an infinite set and let P be the finite-cofinite algebra of \ . Find the Dedekind–MacNeille completion of P. 40. Describe conditions on T that characterize the statement g − DMÐT Ñ. 41. Prove that a poset T is a complete lattice if and only if every subset E of T for which E œ E?j is a principal ideal. 42. Let P be a lattice. Prove that an ideal M ü P is DM-closed if and only if it is the intersection of principal ideals. 43. Let T be a poset. a) Suppose that 5À T Ä U is an order embedding and that 5ÐT Ñ is both join-dense and meet-dense in U. Show that there is an order embedding 7 À U Ä DMÐT Ñ such that 7 ‰ 5 œ 9Æ . b) Prove that if U is a complete lattice then 7 is an isomorphism. 44. Prove that if P is a lattice with no infinite chains (ACC and DCC) then P is isomorphic to the DM completion of ] ÐPÑ ∪ `ÐPÑ. MacNeille Extensions Let 0 À P Ä Q be a monotone function between lattices P and Q , thought of as subsets of their DM-completions. Define the lower MacNeille extension 0 ð À DMÐPÑ Ä DMÐQ Ñ of 0 by 0 ð ÐBÑ œ 2Ö0 Ð+Ñ ± + − Pß + Ÿ B× and the upper MacNeille extension 0 ñ À DMÐPÑ Ä DMÐQ Ñ of 0 by 0 ñ ÐBÑ œ 4Ö0 Ð+Ñ ± + − Pß B Ÿ +× The map 0 is called smooth if 0 ð œ 0 ñ . 45. Show that a) 0 ð and 0 ñ are extensions of 0 b) 0 ð ÐBÑ Ÿ 0 ñ ÐBÑ for all B − P c) 0 ð and 0 ñ are monotone. 46. Show that the join map 1À P ‚ P Ä P is smooth and that 1ð œ 1ñ is the join in DMÐPÑ. A dual statement holds for the meet. 47. Let P be the finite-cofinite algebra of and let # œ Ö!ß "× be the twoelement lattice. Let 0 À P Ä # be defined by Lattices 0 ÐEÑ œ œ " ! 93 if E is cofinite if E is finite Find the Dedekind–MacNeille completion of P. Find the extensions 0 ð and 0 ñ of 0 . Show that 0 is a lattice homomorphism but that 0 ð does not preserve joins and 0 ñ does not preserve meets. 48. Let P be a distributive lattice P containing two elements B and D for which Pw œ DMÐPÑ has a sublattice isomorphic to R& (see Figure 11.4) and is therefore not distributive. Let 0 À P Ä P be defined by 0 Ð?Ñ œ D ” ? a) Show that 0 preserves meets. b) Find 0 ð and show that it does not preserve meets. Galois Connections 49. Show that a pair of maps ÐCß HÑ is a Galois connection for ÐT ß UÑ if and only if for all : − T and ; − U, C: ; Í : Ÿ H; 50. Let T be a poset and let T M be the set of principal ideals and T J be the set of principal filters in T , each of which is ordered by set inclusion. Show that the maps CÐ Æ BÑ È Å B and =À Ð Å BÑ È Æ B form a Galois connection. Chapter 4 Modular and Distributive Lattices In this chapter, we examine modularity and distributivity in more detail. Quadrilaterals At various times throughout the remainder of the book, we will have use for the following concept. Definition Let P be a lattice. As shown on the left in Figure 4.1, a quadrilateral in P is a set consisting of two elements +ß , − P, together with their meet and join. We write this in the form of a %-tuple U œ Ð+ • ,ß +ß ,ß + ” ,Ñ A quadrilateral U is degenerate if U has fewer than % distinct elements (in which case, U has either " or # elements). a∨b β∨x b a α∨x β a∧b α Figure 4.1 Note that the edges in Figure 4.1 are not necessarily meant to indicate the covering relation. Also, a quadrilateral U is degenerate if and only if + and , are comparable. The Definitions For convenience, we repeat the definitions. Definition S. Roman (ed.), Lattices and Ordered Sets, doi: 10.1007/978-0-387-78901-9_4 , © Steven Roman 2008 95 96 Lattices and Ordered Sets 1) A lattice P is modular if it satisfies the modular law: For all +ß ,ß - − P, + - Ê + • Ð, ” -Ñ œ Ð+ • ,Ñ ” - 2) A lattice P is distributive if it satisfies the distributive laws: For all +ß ,ß - − P, + • Ð, ” -Ñ œ Ð+ • ,Ñ ” Ð+ • -Ñ + ” Ð, • -Ñ œ Ð+ ” ,Ñ • Ð+ ” -Ñ We leave it to the reader to verify that the modular law is self-dual. It follows that a lattice is modular if and only if its dual is modular. Theorem 4.1 If either of the distributive laws holds for all elements of a lattice P, then so does the other. Proof. Suppose that the first distributive law holds. Then applying it to the right side of the second distributive law and using absorption gives Ð+ ” ,Ñ • Ð+ ” -Ñ œ ÒÐ+ ” ,Ñ • +Ó ” ÒÐ+ ” ,Ñ • -Ó œ + ” ÒÐ+ ” ,Ñ • -Ó œ + ” ÒÐ+ • -Ñ ” Ð, • -ÑÓ œ + ” Ð, • -Ñ which shows that the second law holds. Note that the distributive laws are dual to one another and so a lattice is distributive if and only if its dual lattice is distributive. It is clear that a distributive lattice is modular, but the converse is false, as we will see. The usual definitions of modular and distributive lattices given above are not as “sharp” as possible. For example, a lattice is distributive if and only if either of the following inequalities holds: + • Ð, ” -Ñ Ÿ Ð+ • ,Ñ ” Ð+ • -Ñ + ” Ð, • -Ñ Ð+ ” ,Ñ • Ð+ ” -Ñ since the reverse inequalities hold in all lattices. Similarly, a lattice is modular if and only if +- Ê + • Ð, ” -Ñ Ÿ Ð+ • ,Ñ ” - since the reverse inequality holds in all lattices. Examples Let us consider some examples. Example 4.2 The lattice fÐKÑ of subgroups of a group need not be modular. The alternating group E% provides an example: Let Modular and Distributive Lattices + œ ØÐ" #ÑÐ$ %Ñß Ð" $ÑÐ# %ÑÙß , œ ØÐ" # $ÑÙ 97 and - œ ØÐ" #ÑÐ$ %ÑÙ On the other hand, it is easy to see that the sublattice of normal subgroups a ÐKÑ of a group K is modular. In a ÐKÑ, meet is intersection and join is the set product, that is, if Eß F − a ÐKÑ, then E•F œE∩F and E ” F œ EF and the modular law is E G Ê E ∩ FG œ ÐE ∩ FÑG a fact known in group theory as Dedekind's law. It can be shown that fÐKÑ is distributive if and only if K is locally cyclic (that is, any finite nonempty subset of K generates a cyclic subgroup). The proof is a bit complex (see Hall, The Theory of Groups [31]). Thus, if K is finite, then fÐKÑ is distributive if and only if K is cyclic. The proof that a finite cyclic group has a distributive subgroup lattice is not hard and is left as an exercise. Example 4.3 The lattice of subspaces of a vector space of dimension at least # is modular but not distributive. Example 4.4 1) Any sublattice of a power set lattice kÐWÑ is distributive. 2) Any chain is distributive. 3) The lattice Ðß ± Ñ is distributive. 4) The lattice of subgroups of ™ is distributive. 5) The lattices Q$ and R& of Figure 4.2 are not distributive. Of course, distributivity extends to finite joins and meets, but not necessarily to infinite ones. Example 4.5 The complete lattice Ðß ± Ñ is distributive. However, let E be the set of odd natural numbers. Then 1E œ ! and so # • Š2E‹ œ # On the other hand, # • + œ " for all odd numbers and so 2 Ð# • +Ñ œ " Á ! +−E This example indicates one way in which arbitrary distributive lattices differ fundamentally from power sets: In a power set, for example, meet distributes 98 Lattices and Ordered Sets over infinite joins, that is, E ∩ Š.F3 ‹ œ .ÐE ∩ F3 Ñ Inheritance by \ ÐPÑ We have seen (Theorem 3.35) that if P is a lattice, then so is the set \ ÐPÑ of all ideals of P. Moreover, the down map 9Æ À P Ä \ ÐPÑ is a lattice embedding. Therefore, if \ ÐPÑ is modular, then so is P and if \ ÐPÑ is distributive, then so is P. The converse is also true. Theorem 4.6 Let P be a lattice. 1) P is modular if and only if \ ÐPÑ is modular. 2) P is distributive if and only if \ ÐPÑ is distributive. Proof. For part 1), suppose that E ª G . We must show that E ∩ ÐF ” GÑ © ÐE ∩ FÑ ” G If B − E ∩ ÐF ” GÑ, then B œ + − E and B Ÿ , ” - for , − F and - − G . Thus, since + ” - − E, we have B Ÿ + • Ð, ” -Ñ Ÿ Ð+ ” -Ñ • Ð, ” -Ñ œ ÒÐ+ ” -Ñ • ,Ó ” - − ÐE ∩ FÑ ” G and so \ ÐPÑ is modular. For part 2), if P is distributive, we must show that ÐE ” FÑ ∩ G © ÐE ∩ GÑ ” ÐF ∩ GÑ If B − ÐE ” FÑ ∩ G , then B Ÿ + ” , for some + − E and , − F and B œ - − G , whence B Ÿ Ð+ ” ,Ñ • - œ Ð+ • -Ñ ” Ð, • -Ñ − ÐE ∩ GÑ ” ÐF ∩ GÑ for + − E, , − F and - − G . Hence, \ ÐPÑ is distributive. Characterizations There are several useful ways to characterize modularity and distributivity. Let us look at some of the most important ones. Characterizations by Forbidden Sublattice Modularity and distributivity can be characterized by the absence of certain sublattices. As we shall see, these characterizations can be extremely useful in showing that certain properties imply modularity or distributivity. The offending lattices are shown in Figure 4.2. From now on, we will reserve the notations Q$ and R& for these lattices. The lattice Q$ is called a diamond and the lattice R& is called a pentagon. Note that some authors use the notation Q& rather than Q$ . (Grätzer [24] uses Q$ but Birkhoff [5] uses Q& .) Modular and Distributive Lattices b x 99 b y x z y z a M3 a N5 Figure 4.2 Theorem 4.7 1) A lattice P is modular if and only if it does not have a sublattice isomorphic to R& . 2) A lattice P is distributive if and only if it does not have a sublattice isomorphic to either R& or Q$ . Proof. For part 1), it is easy to see that R& is not modular and so any lattice with a sublattice isomorphic to R& is not modular. Conversely, if P is not modular, then there exist +ß ,ß - − P such that + - and B ³ + • Ð, ” -Ñ D ³ Ð+ • ,Ñ ” - To see that W œ ÖB, ,ß Dß B • ,ß B ” ,× is an R& sublattice of P, we have D B and D ” , œ Ð+ • ,Ñ ” - ” , œ - ” , B and so D ” , B ” , . But the reverse inequality is clear and so D ” , œ B ” , . Also, B • , œ + • Ð, ” -Ñ • , œ + • , Ÿ D and so B • , Ÿ D • , . But the reverse inequality is clear and so D • , œ B • , . Finally, to show that B ² , and D ² , , we have the following: 1) 2) 3) 4) If , Ÿ B, then , Ÿ + and since - Ÿ +, it follows that B œ D , a contradiction. If D Ÿ , , then - Ÿ , and since - Ÿ +, it follows that B œ D , a contradiction. If B , , then D B , , which is false by 2). If , D implies , D B, which is false by 1). Thus, , is parallel to both B and D and W is R& . For part 2), if P contains one of Q$ or R& , then P is not distributive since these lattices are not distributive. Conversely, suppose that P is not distributive. If P is not modular, then it has an embedded R& , so we may assume that P is modular. 100 Lattices and Ordered Sets Since P is not distributive, there exist +ß ,ß - − P such that B ³ + • Ð, ” -Ñ D ³ Ð+ • ,Ñ ” Ð+ • -Ñ We use this inequality to rule out certain relationships between +, , and - . In particular, we must have the following: 1) 2) 3) 4) +ß ,ß - must be distinct +Ÿ y - and + Ÿ y , + - and + , cannot both hold , ² -. Thus, we are left with two possibilities: 5) + - , + ² , , , ² - (or symmetrically, + ² - , + , , , ² - ) 6) + ² - , + ² , , , ² - . In case 5), we have + - and the modularity of P implies that B œ + • Ð, ” -Ñ œ Ð+ • ,Ñ ” - œ D which is false. Thus, 5) can be eliminated as well. Thus, we are left with the case in which +, , and - are pairwise incomparable. We clain that the elements : ³ Ð+ • ,Ñ ” Ð+ • -Ñ ” Ð, • -Ñ ; ³ Ð+ ” ,Ñ • Ð+ ” -Ñ • Ð, ” -Ñ ? ³ Ð+ • ;Ñ ” : @ ³ Ð, • ;Ñ ” : A ³ Ð- • ;Ñ ” : form an Q$ sublattice W of P, where ?, @ and A are the incomparable elements of W , : is the smallest element and ; is the largest element. We need to show that these elements are distinct and that : œ?•@œ?•Aœ@•A and ; œ?”@œ?”Aœ@”A It is clear that : Ÿ ; . To see that : Á ; , note that + • ; œ + • Ð, ” -Ñ œ B and, using the modular law, Modular and Distributive Lattices 101 + • : œ + • ÒÐ+ • ,Ñ ” Ð+ • -Ñ ” Ð, • -ÑÓ œ Ò+ • Ð, • -ÑÓ ” ÒÐ+ • ,Ñ ” Ð+ • -ÑÓ œ Ð+ • ,Ñ ” Ð+ • -Ñ œD and since B D , we must have : ; . To show that ? • @ œ :, we have ? • @ œ ÒÐ+ • ;Ñ ” :Ó • ÒÐ, • ;Ñ ” :Ó and since Ð+ • ;Ñ ” : : and ; :, multiple applications of the modular law give ? • @ œ ÒÒÐ+ • ;Ñ ” :Ó • Ð, • ;ÑÓ ” : œ ÒÒ; • Ð+ ” :ÑÓ • Ð, • ;ÑÓ ” : œ ÒÐ, ” :Ñ • Ð+ • ;ÑÓ ” : Now, , ” : œ , ” Ð+ • -Ñ and + • ; œ + • Ð, ” -Ñ and so a bit of rearranging gives ? • @ œ ÖÒ, ” Ð+ • -ÑÓ • Ò+ • Ð, ” -ÑÓ× ” : œ Ö+ • ÒÐ, ” -Ñ • ÒÐ+ • -Ñ ” ,ÓÓ× ” : Another application of the modular law using the fact that , ” - , and then yet another using the fact that + + • - gives ? • @ œ Ö+ • ÒÒÐ, ” -Ñ • Ð+ • -ÑÓ ” ,Ó× ” : œ Ö+ • ÒÐ+ • -Ñ ” ,Ó× ” : œ ÖÐ+ • ,Ñ ” Ð+ • -Ñ× ” : œ: In a similar way, it can be shown that the other equalities hold. As to uniqueness, if ? œ @, for example, then : œ ? • @ œ ? ” @ œ ; , which is false. Thus, we see that ?, @ and A are distinct. If ? œ ; , then ? œ ? ” @ and so @ Ÿ ? and similarly, A Ÿ ?, whence @ œ @ • ? œ A • ? œ A, which is false. In this way, we see that all five elements are distinct. Characterization by Relative Complements The characterizations by forbidden sublattice can be rephrased in terms of relative complements. Indeed, saying that R& is a sublattice of P is equivalent to saying that there are elements Bß Cß Dß +ß , in P for which B D and Bß D − CÒ+ß,Ó . Thus, R& is not a sublattice of P if and only if no set CÒ+ß,Ó of relative complements has two distinct comparable elements. 102 Lattices and Ordered Sets Similarly, saying that either one of Q$ or R& is a sublattice of P is equivalent to saying that there are elements Bß Cß Dß +ß , in P for which Bß D − CÒ+ß,Ó , where either B D or B ² D . Hence, neither Q$ nor R& is a sublattice of P if and only if the sets CÒ+ß,Ó are either empty or singletons. Theorem 4.8 1) A lattice P is modular if and only if every set +Ò?ß@Ó of relative complements is either empty or is an antichain. 2) A lattice P is distributive if and only if relative complements in P, when they exist, are unique or equivalently, if and only if every element of P is uniquely determined by its meet and join with any other element, that is, B•, œB•œB ” , œ B ” - Ê ,œ- In particular, uniquely relatively complemented lattices are distributive. Thus, uniquely relatively complemented lattices are distributive. However, the converse fails: Distributive lattices need not be complemented, let alone relatively complemented, as witnessed by any chain with more than two elements, or by the lattice in Figure 4.3. 1 z x y 0 Figure 4.3 Unique Complementation and Distributivity In 1904, E. V. Huntington [34] conjectured that any uniquely complemented lattice was distributive. For over 40 years this remained one of the main problems of lattice theory. It was not until 1945 that Dilworth [16], in a rather grueling 32 pages, showed that every lattice is a sublattice of a uniquely complemented lattice. Hence, unique complementation cannot imply distributivity, since otherwise every lattice would be distributive. Much work has been done on the question of what additional properties are required of a uniquely complemented lattice in order to imply distributivity. We will discuss this question in some detail when we discuss Boolean algebras in a later chapter. (A complemented distributive lattice is a Boolean algebra.) As a Modular and Distributive Lattices 103 preview, for a uniquely complemented lattice P, any one of the following conditions implies distributivity: 1) 2) 3) 4) P is modular P satisfies de Morgan's laws P is atomic P has finite width. In view of these conditions, it is not surprising that it is hard to find a “natural” example of a nondistributive, uniquely complemented lattice. Characterization Using the Join and Meet Maps The modular and distributive laws can also be described in terms of the join maps 4+ À P Ä P and the meet maps 7+ À P Ä P, defined by 4+ ÐBÑ œ B ” + and 7+ ÐBÑ œ B • + Distributivity has a simple characterization in terms of these maps. Theorem 4.9 Let P be a lattice. 1) P is distributive if and only if the join map 4+ is a lattice homomorphism for all + − P. 2) Dually, P is distributive if and only if the meet map 7+ is a lattice homomorphism for all + − P. Proof. It is easy to see that 4+ preserves joins. As to preservation of meets, the equation 4+ ÐB • CÑ œ 4+ ÐBÑ • 4+ ÐCÑ is none other than the distributive law + ” ÐB • CÑ œ Ð+ ” BÑ • Ð+ ” CÑ As to modularity, with reference to the quadrilateral in Figure 4.4, if P is modular, then for any +ß , − P and B − Ò+ • ,ß +Ó, 7+ ‰ 4, ÐBÑ œ ÐB ” ,Ñ • + œ B ” Ð, • +Ñ œ B and so 7+ ‰ 4, œ + is the identity map on the interval Ò+ • ,ß +Ó. 104 Lattices and Ordered Sets a∨b jb a ma b a∧b Figure 4.4 Conversely, if P is not modular, then P contains a sublattice R& , where D − ÒB • Cß BÓ. But 7B ‰ 4C ÐDÑ œ ÐD ” CÑ • B œ B Á D Hence, 7B ‰ 4C Á + . Thus, P is modular if and only if 7+ ‰ 4, œ + on Ò+ • ,ß +Ó, for all +ß , − P. Theorem 4.10 Let P be a lattice and let and 7+ À Ò,ß + ” ,Ó Ä Ò+ • ,ß +Ó 4, À Ò+ • ,ß +Ó Ä Ò,ß + ” ,Ó be the join and meet maps, restricted to the opposite sides of the quadrilateral in Figure 4.4. The following are equivalent: 1) P is modular. 2) For all +ß , − P, 7+ ‰ 4, œ + on Ò+ • ,ß +Ó. 3) Dually, for all +ß , − P, 4, ‰ 7+ œ + on Ò,ß + ” ,Ó. 4) For all +ß , − P, the restricted maps 7+ and 4, are inverse lattice isomorphisms on these intervals. Proof. We have seen that 1)–3) are equivalent and since 7+ and 4, are orderpreserving, 2) and 3) imply that these maps are inverse order (lattice) isomorphisms and so 4) holds. If 4) holds, then clearly 2) holds. Modularity and Semimodularity Recall that + ö , means that + covers , and + « , means that + ö , or + œ , . It is too much to expect that the join map 4B À P Ä P preserves covering, that is, that +ö, Ê +”Bö,”B for if B , , then + ” B œ , ” B œ B. On the other hand, it is possible that the Modular and Distributive Lattices 105 join map weakly preserves covering, that is, +ö, Ê +”B«,”B Indeed, this happens in any modular lattice P. To see this, suppose that + ö , and that + ” B Ÿ ? Ÿ , ” B. Since ? + ” B, the modular law gives ? œ ? • Ð, ” Ð+ ” BÑÑ œ Ð? • ,Ñ ” Ð+ ” BÑ But since + ö , and + Ÿ ? • , Ÿ , , it follows that ? • , œ + or ? • , œ , . Hence, ? œ + ” B or ? œ , ” B, which shows that + ” B « , ” B. The weak preservation of covering is equivalent to the condition α•" öα Ê (4.11) " öα”" for all αß " − P. To see that (4.11) is implied by the weak preservation of covering, let α • " ö α and set + œ α • " , , œ α and B œ , . Then + ” B « , ” B, where + ” B œ Ðα • " Ñ ” " œ " and , ” B œ α ” " and so " « α ” " . But " œ α ” " implies that α • " œ α, which contradicts the fact that α • " ö α. Hence, " ö α ” " . Conversely, assume that (4.11) holds and that + ö , . Then + Ÿ , • Ð+ ” BÑ Ÿ , and so , • Ð+ ” BÑ œ + or , • Ð+ ” BÑ œ , . But in the latter case, , Ÿ + ” B and so , ” B œ + ” B. In the case , • Ð+ ” BÑ œ +, we have , • Ð+ ” BÑ ö , and so (4.11) implies that + ” B ö , ” Ð+ ” BÑ œ , ” B as desired. The two equivalent conditions are shown in Figure 4.5 and have a name (see Grätzer [24]). α∨β b∨x a∨x α b β α∧β a Figure 4.5 Definition Let P be a lattice. 1) As shown in Figure 4.5, the upper covering condition on P is the condition that the join map 4B À P Ä P weakly preserves covering, that is, 106 Lattices and Ordered Sets +ö, Ê +”B«,”B for all +ß ,ß B − P, or equivalently +•, ö+ Ê , ö+”, for all +ß , − P. A lattice that has the upper covering condition is called an upper semimodular lattice. 2) Dually, the lower covering condition on P is the condition that the meet map 7B À P Ä P weakly preserves covering, that is, +ö, Ê +•B«,•B for all +ß ,ß B − P, or equivalently , ö+”, Ê +•, ö+ for all +ß , − P. A lattice that has the lower covering condition is called a lower semimodular lattice. Theorem 4.12 Every modular lattice is upper semimodular and lower semimodular. Example 4.13 The lattice in Figure 4.6 is upper semimodular but not modular. Figure 4.6 For lattices with no infinite chains, the converse of Theorem 4.12 holds. To explore this, we need a preliminary result. Theorem 4.14 Let P be a lattice with no infinite chains and let 2À P Ä be the height function. 1) If P is upper semimodular, then P satisfies the Jordan–Dedekind chain condition, that is, any two maximal chains from + to , have the same length. 2) Dually, if P is lower semimodular, then P satisfies the Jordan–Dedekind chain condition. 3) P is upper semimodular if and only if 2Ð+Ñ 2Ð,Ñ 2Ð+ • ,Ñ 2Ð+ ” ,Ñ Modular and Distributive Lattices 107 4) Dually, P is lower semimodular if and only if 2Ð+Ñ 2Ð,Ñ Ÿ 2Ð+ • ,Ñ 2Ð+ ” ,Ñ Proof. Note first that since P has no infinite chains, a chain from + to , is maximal if and only if each step in the chain is a covering. We prove part 1) by induction on 8 that if P has a maximal chain of length 8 from + to , , then all maximal chains from + to , have length 8. If 8 œ ", then + ö , . But if + ö ?" ö â ö ?7 œ , is a maximal chain from + to , , then + ?" Ÿ , and so ?" œ , and this chain also has length ". Assume that the result is true for all chains of length less than 8. With reference to Figure 4.7, b C x2 E D x1∨y1 x1 y2 y1 a Figure 4.7 let V œ Ö+ ö B" ö B# ö â ö B8" ö ,× and W œ Ö+ ö C" ö C# ö â ö C7" ö ,× be maximal chains from + to , . If B" œ C" , then the induction hypothesis implies that 8 " œ 7 ", so assume that B" Á C" . Since + ö B" and + ö C" , the upper covering condition implies that B" ö B" ” C" and C" ö B" ” C" . Let X be a maximal chain from B" ” C" to , . Prefixing X with B" gives a maximal chain from B" to , and so the induction hypothesis implies that lenÐX Ñ œ 8 ". Similarly, lenÐX Ñ œ 7 " and so 7 œ 8. Part 3) can be rephrased by saying that P is upper semimodular if and only if .Ð+ • ,ß +Ñ .Ð,ß + ” ,Ñ 108 Lattices and Ordered Sets for all +ß , − P, where .ÐBß CÑ is the length of any maximal chain from B to C. But, as shown in Figure 4.8, a∨b a ... ... x2∨b x1∨b b x2 x1 a∧b Figure 4.8 if + • , ö B" ö B# ö â ö B8" ö + is a maximal chain from + • , to + of length 8, then the upper covering condition gives , « B" ” , « B# ” , « â « B8" ” , « + ” , and so there is a maximal chain from , to + ” , of length at most 8. Hence, .Ð+ • ,ß +Ñ .Ð,ß + ” ,Ñ Conversely, if 3) holds and + • , ö +, then " œ .Ð+ • ,ß +Ñ .Ð,ß + ” ,Ñ But , Á + • , and so .Ð,ß + ” ,Ñ œ ", that is, , ö + ” , and so P is upper semimodular. Theorem 4.15 Let P be a lattice with no infinite chains. The following are equivalent: 1) P is modular 2) P is upper and lower semimodular 3) The height function 2 satisfies 2Ð+Ñ 2Ð,Ñ œ 2Ð+ • ,Ñ 2Ð+ ” ,Ñ or, equivalently, .Ð+ • ,ß +Ñ œ .Ð,ß + ” ,Ñ for all +ß , − P. Proof. We have seen that 1) implies 2) and that 2) implies 3). Suppose that 3) holds. If P is not modular, then it contains a sublattice R& , as shown in Figure Modular and Distributive Lattices 109 4.2. Hence, 2ÐBÑ 2ÐCÑ œ 2ÐB ” CÑ 2ÐB • CÑ and 2ÐDÑ 2ÐCÑ œ 2ÐD ” CÑ 2ÐD • CÑ But the right-hand sides of these two equations are equal and so 2ÐBÑ œ 2ÐDÑ, which is false. The definition of upper and lower covering condition is that of Grätzer [24]. Birkhoff [5] employs a somewhat stronger form of covering condition that is equivalent when P has no infinite chains. We will refer to it by the nonstandard name double upper covering condition. Definition Let P be a lattice. With reference to Figure 4.9, we make the following definitions. a∨b a∨b b a a a∧b b a∧b Figure 4.9 1) The double upper covering condition is + • , ö +ß + • , ö , Ê + ö + ” ,ß , ö + ” , 2) Dually, the double lower covering condition is + ö + ” ,ß , ö + ” , Ê + • , ö +ß + • , ö , It is clear that the upper covering condition implies the double upper covering condition. The converse holds if P has no infinite chains. Theorem 4.16 A lattice with no infinite chains has the upper covering condition if and only if it has the double upper covering condition. A similar statement holds for lower in place of upper. Proof. Suppose that the double upper covering condition holds and that + ö , . We show that + ” B « , ” B for any B − P. We may assume that + ” B Á , ” B and so ,  Ò+ß + ” BÓ. Consider a maximal chain + œ ?! ö ?" ö â ö ?8" ö ?8 œ + ” B from + to + ” B, as shown in Figure 4.10. 110 Lattices and Ordered Sets b∨un=b∨x b∨uk+1 un=a∨x b∨uk uk+1 b∨u1 uk b∨u0=b u1 u0=a Figure 4.10 Since ?! ö , ” ?! and ?! ö ?" , the double upper covering condition implies that ?" ö , ” ? " and , ” ?! ö , ” ?" More generally, we have for all 5 œ !ß á ß 8 ", ?5 ö , ” ?5 and ?5 ö ?5" Ê ?5" ö , ” ?5" and , ” ?5 ö , ” ?5" Hence, ?8 ö , ” ?8 , that is, + ” B ö , ” B. For more on semimodular lattices, we refer the reader to Grätzer [24] or Stern [58]. Partition Lattices and Representations We begin with the definition of a partition. Definition A partition of a nonempty set \ is a collection ÖE3 ± 3 − M× of nonempty subsets of W , called the blocks of the partition, for which 1) E3 ∩ E4 œ g for all 3 Á 4 2) W œ -E3 . A block is trivial if it has size ". It is well known that there is a bijection between the family EquÐ\Ñ of equivalence relations on a nonempty set \ and the family PartÐ\Ñ of partitions of W . When we write EquÐ\Ñ or PartÐ\Ñ, it is with the understanding that \ is a nonempty set. Thinking of equivalence relations as subsets of \ # , it is clear that EquÐ\Ñ is a poset under set inclusion. For PartÐ\Ñ, we have the following order. Modular and Distributive Lattices 111 Definition Let T and U be partitions of a nonempty set \ . Then T is a refinement of U, written T Ÿ U, if each block of U is a union of one or more blocks of T . If T Ÿ U, we say that T is finer than U and U is coarser than T . Theorem 4.17 Let \ be a nonempty set. The bijection CÀ EquÐ\Ñ Ä PartÐ\Ñ that maps an equivalence relation ) to the corresponding partition T) is an order isomorphism, that is, )©5 Í T) Ÿ T 5 In view of this theorem, we will describe our results in terms of either partitions or equivalence relations, whichever seems clearer at the time. Since EquÐ\Ñ is an intersection-structure, it is a complete lattice. To describe the join of equivalence relations, we use the following concept. Definition Let Y be a family of binary relations on a nonempty set \ . If +ß , − \ , then a witness sequence from + to , is a sequence + œ +" ß +# ß á ß +8" ß +8 œ , in \ for which )" )8" )# + œ +" ´ +# ´ +$ ´ â ´ +8" ´ +8 œ , where )3 − Y . If there is a witness sequence from + to , , we say that , is reachable from + using Y . The length of a witness sequence is the number of terms in the sequence, not including the first and last. Thus, + ´ B ´ , has length ". Theorem 4.18 Let \ be a nonempty set. The poset EquÐ\Ñ is an intersectionstructure and therefore a complete lattice. Let Y œ Ö)3 ± 3 − M× be a family of equivalence relations on \ , with corresponding partitions c œ ÖT3 ± 3 − M×, where T3 œ T)3 . 1) The meet is given by 4Y œ ,Y and so +Š4Y ‹, if +), for all ) − Y The meet of the family c can be described as the coarsest common refinement of c . Thus, the blocks of 3c are the largest subsets of \ that fit inside a single block from each member of c . 112 Lattices and Ordered Sets 2) The join ) œ 2Y of Y is the transitive closure of the union -Y . Specifically, +), if and only if there is a witness sequence )3 " )38" )3 # + œ +" ´ +# ´ +$ ´ â ´ +8" ´ +8 œ , from + to , using Y . The join of c can be described as follows: Two elements +ß , − \ are equivalent under the join if there is a finite sequence of “jumps” starting with + and ending with , , where each jump must lie within a single block of some member of c . Proof. We leave the proof that EquÐ\Ñ is an intersection-structure to the reader. For part 2), let ) œ 1Y . The reachable relation V is easily seen to be an equivalence relation. Moreover, since )3 © ), it follows that +V, implies +), , that is, V © ) . But )5 © V for all 5 and so ) © V . Thus, ) œ V . We leave proof of the following for the exercises. Theorem 4.19 Let \ be a nonempty set. 1) The atoms in PartÐ\Ñ are the partitions with only one nontrivial block, which has size #. 2) For the covering relation in PartÐ\Ñ, we have 5 ö ) if and only if ) is obtained from 5 by taking the union of exactly two distinct blocks of 5 . 3) PartÐ\Ñ is atomistic and strongly atomic. 4) PartÐ\Ñ is semimodular. 5) PartÐ\Ñ is relatively complemented. Representations Cayley's theorem of group theory says that any group can be embedded in (is isomorphic to a subgroup of) a permutation group. Thus, permutation groups are universal, in the sense that they “contain” all groups. Partition lattices play the analogous role for lattices. Definition Let P be a lattice. 1) A representation of a lattice P is a lattice embedding AÀ P ä PartÐ\Ñ for some nonempty set \ . 2) A representation A has a) type " if for all +ß , − P and Bß C − \ , BÐA+ ” A,ÑC for some D − \ . Ê BÐA+ÑDÐA,ÑC Modular and Distributive Lattices 113 b) type # if for all +ß , − P and Bß C − \ , BÐA+ ” A,ÑC c) Ê BÐA+ÑD" ÐA,ÑD# ÐA+ÑC for some D" ß D# − \ . type $ if for all +ß , − P and Bß C − \ , BÐA+ ” A,ÑC Ê BÐA+ÑD" ÐA,ÑD# ÐA+ÑD$ ÐA,ÑC for some D" ß D# ß D$ − \ . Note that the definition of type and the commutativity of join imply that there is a witness sequence from B to C that starts with A, as well. In 1946, P. M. Whitman [67] proved that every lattice has a representation. In 1953, Bjarni Jónsson [37] proved that every lattice has a type $ representation. We follow the proof more or less as given in Grätzer [24] and for this proof, we require two definitions. Definition A meet representation of a lattice P is a meet-preserving bijection from P to PartÐ\Ñ, for some nonempty set \ . Definition Let P be a lattice with ! and let \ be a nonempty set. A function $À \ ‚ \ Ä P is a distance function if 1) $ is surjective. 2) $ is symmetric, that is, $ ÐBß CÑ œ $ ÐCß BÑ for all Bß C − \ . 3) $ is normalized, that is, $ÐBß BÑ œ ! for all B − \ . 4) $ satisfies the triangle inequality, that is, $ ÐBß DÑ Ÿ $ ÐBß CÑ ” $ ÐCß DÑ for all Bß Cß D − \ . For example, the function $À P ‚ P Ä P given by $Ð+ß ,Ñ œ œ +”, ! if + Á , if + œ , is a distance function. Theorem 4.20 Let P be a lattice. A distance function $ À \ ‚ \ Ä P defines a meet-representation 114 Lattices and Ordered Sets $À P Ä PartÐ\Ñ of P by (in equivalence relation language) B$ Ð+ÑC if $ ÐBß CÑ Ÿ + for all + − E and Bß C − \ . Proof. It is clear that $ Ð+Ñ is reflexive and symmetric. If B$ Ð+ÑC$ Ð+ÑD , then $ ÐBß CÑ Ÿ + and $ ÐCß DÑ Ÿ + and so the triangle inequality shows that $ ÐBß DÑ Ÿ +. Hence, B$ Ð+ÑD and so $ Ð+Ñ is transitive and therefore an equivalence relation. Now, B$ Ð+ • ,ÑC Í B ” C Ÿ + • , Í B ” C Ÿ + and B ” C Ÿ , Í B$ Ð+ÑC and B$ Ð,ÑC Í BÒ$ Ð+Ñ ∩ $ Ð,ÑÓC and so $ preserves meets. To see that $ is injective, if $ Ð+Ñ œ $ Ð,Ñ, then $ ÐBß CÑ Ÿ + if and only if $ ÐBß CÑ Ÿ , . However, if + Á , , we may assume that +Ÿ y , and since $ is surjective, there are Bß C − \ for which $ ÐBß CÑ œ +, but then $ ÐBß CÑ Ÿ y , . Hence, $ is a meet representation of P. Definition Let \ © ] and let $ À \ ‚ \ Ä P and %À ] ‚ ] Ä P be distance functions. Then % extends $ if %l\‚\ œ $ . It is clear that if % extends $ , then $ Ÿ % on \ . Theorem 4.21 (Jónsson [37]) 1) Every lattice P has a type $ representation. 2) A lattice P has a type # representation if and only if it is modular. Proof. We first prove that any lattice P with a type # representation is modular. For convenience, let us identify an element + − P with its image A+ under the representation AÀ P Ä PartÐ\Ñ. We must show that for +ß ,ß - − P with + - , + • Ð, ” -Ñ Ÿ Ð+ • ,Ñ ” that is, for any Bß C − \ , BÒ+ • Ð, ” -ÑÓC Ÿ BÒÐ+ • ,Ñ ” -ÓC Now, the left side implies that B+C and BÐ, ” -ÑC and since the representation has type #, there are D" ß D# − \ for which B-D" ,D# -C Thus, D" -B+C-D# and so - Ÿ + implies that D" +D# and since D" ,D# , it follows that Modular and Distributive Lattices 115 D" Ð+ • ,ÑD# , whence BÒÐ+ • ,Ñ ” -ÓC , which is the right-hand side of the modular inequality. Hence, P is modular. For part 1) and the remaining direction of part 2), suppose that $À \ ‚ \ Ä P is a distance function, with corresponding meet representation $À P Ä PartÐ\Ñ. Suppose further that ?$ Ð+ ” ,Ñ@ for some +ß , − P and ?ß @ − \ , that is, $Ð?ß @Ñ Ÿ + ” , Then it may fail to hold that ?Ò$ Ð+Ñ ” $ Ð,ÑÓ@, but we can adjoin some elements to \ ($ elements in general and # elements when P is modular) and extend $ so that this is the case, with a witness sequence from ? to @ of length $ in general and of length # when P is modular. This is done as follows. For any P (modular or not), let D" , D# and D$ be symbols not in \ and let \ ‡ œ \ ∪ ÖD" ß D# ß D$ × Extend $ to $ ‡ by setting $ ‡ ÐD" ß D# Ñ œ , $ ‡ ÐD# ß D$ Ñ œ + $ ‡ ÐD" ß D$ Ñ œ + ” , and for all B − \ , $ ‡ ÐBß D" Ñ œ $ ÐBß ?Ñ ” + $ ‡ ÐBß D# Ñ œ $ ÐBß ?Ñ ” + ” , $ ‡ ÐBß D$ Ñ œ $ ÐBß @Ñ ” , and require also that $ ‡ be symmetric and normalized. These extensions may seem arbitrary at first, but they are designed so that the resulting function $ ‡ À \ ‡ ‚ \ ‡ Ä P will satisfy the triangle inequality and therefore be a distance function. In particular, by matching + and , as above, the right side of the triangle inequality $ ‡ ÐBß D4 Ñ Ÿ $ ‡ ÐBß D3 Ñ ” $ ‡ ÐD3 ß D4 Ñ (4.22) has the form $ÐBß ? or @Ñ ” + ” , whereas the left side has the form $ÐBß ? or @Ñ ” where - Ÿ + ” , . Moreover, since $ satisfies the triangle inequality, we have $ ÐBß ?Ñ Ÿ $ ÐBß @Ñ ” $ Ð@ß ?Ñ Ÿ $ ÐBß @Ñ ” + ” , 116 Lattices and Ordered Sets and similarly, $ ÐBß @Ñ Ÿ $ ÐBß ?Ñ ” + ” , Hence, it is clear that (4.22) holds. We leave it to the reader to check the other cases of the triangle inequality. Thus, $ ‡ is a distance function. Moreover, the extension also satisfies $ ‡ Ð?ß D" Ñ Ÿ + $ ‡ ÐD" ß D# Ñ Ÿ , $ ‡ ÐD# ß D$ Ñ Ÿ + $ ‡ ÐD$ ß @Ñ Ÿ , that is, ?$ ‡ Ð+ÑD" $ ‡ Ð,ÑD# $ ‡ Ð+ÑD$ $ ‡ Ð,Ñ@ which is a witness sequence from ? to @ under $ ‡ Ð+Ñ ” $ ‡ Ð,Ñ of length $, as promised. Let us refer to Ð\ ‡ ß $ ‡ Ñ as the $-extension of Ð\ß $ Ñ by Ð+ß ,ß ?ß @Ñ and write Ð\ ‡ ß $ ‡ Ñ œ Ð\ß $ Ñ Å Ð+ß ,ß ?ß @Ñ When P is modular, we can accomplish the same goal by adjoining only two new symbols, \ ‡ œ \ ∪ ÖD" ß D# × and extending $ to $ ‡ by setting $ ‡ ÐD" ß D# Ñ œ Ð$ Ð?ß @Ñ ” +Ñ • , $ ‡ ÐBß D" Ñ œ $ ÐBß ?Ñ ” + $ ‡ ÐBß D# Ñ œ $ ÐBß @Ñ ” + for all B − \ and by requiring that $ ‡ be symmetric and normalized. This is also a distance function. As for the only nontrivial case of the triangle inequality, $ ‡ ÐBß D# Ñ Ÿ $ ‡ ÐBß D" Ñ ” $ ‡ ÐD" ß D# Ñ the right-hand side is $ ‡ ÐBß D" Ñ ” $ ‡ ÐD" ß D# Ñ œ $ ÐBß ?Ñ ” + ” ÒÐ$ Ð?ß @Ñ ” +Ñ • ,Ó Since + Ÿ $ Ð?ß @Ñ ” +, the modular law gives œ $ ÐBß ?Ñ ” ÒÐ$ Ð?ß @Ñ ” +Ñ • Ð+ ” ,ÑÓ But $ Ð?ß @Ñ Ÿ + ” , and so $ Ð?ß @Ñ ” + Ÿ + ” , , which reduces this to œ $ ÐBß ?Ñ ” $ Ð?ß @Ñ ” + Modular and Distributive Lattices 117 Finally, the triangle inequality for . gives œ $ ÐBß @Ñ ” + œ $ ‡ ÐBß D# Ñ which is the left-hand side. Thus, $ ‡ is a distance function. Moreover, $ ‡ Ð?ß D" Ñ Ÿ + $ ‡ ÐD" ß D# Ñ Ÿ , $ ‡ ÐD# ß @Ñ Ÿ + that is, ?$ ‡ Ð+ÑD" $ ‡ Ð,ÑD# $ ‡ Ð+Ñ@ which is a witness sequence from ? to @ under $ ‡ Ð+Ñ ” $ ‡ Ð,Ñ of length #, as promised. Let us refer to this pair Ð\ ‡ ß $ ‡ Ñ as the #-extension of Ð\ß $ Ñ by Ð+ß ,ß ?ß @Ñ and also write Ð\ ‡ ß $ ‡ Ñ œ Ð\ß $ Ñ Å Ð+ß ,ß ?ß @Ñ Now, for either case, we need an extension \ of \ and $ of $ so that given any +ß , − P and ?ß @ − \ for which ?$ Ð+ ” ,Ñ@, that is, for which $ Ð?ß @Ñ Ÿ + ” , then ?Ò$ Ð+Ñ ” $ Ð,ÑÓ@, with witness sequence of length $ in general and length # when P is modular. To this end, we first well-order the set of all quadruples Ð+ß ,ß ?ß @Ñ where +ß , − P, ?ß @ − \ and $ Ð?ß @Ñ Ÿ + ” , to form a transfinite sequence ØÐ+α ß ,α ß ?α ß @α Ñ ± α -Ù Then we accumulate extensions. First, let Ð\! ß $! Ñ œ Ð\ß $ Ñ Å Ð+! ß ,! ß ?! ß @! Ñ where Å stands for the $-extension in general or the #-extension when P is modular. For each ordinal " α, suppose we have defined Ð\" ß $" Ñ in such a way that if # " α, then \# © \" and $# © $" . Then the union of the $" 's is a distance function . and we let $ À ’." α \" “ " α " ‚ ’." α \" “ Ä P 118 Lattices and Ordered Sets Ð\α ß $α Ñ œ Š." α \" ß ." α $" ‹ Å Ð+α ß ,α ß ?α ß @α Ñ and \ œ . α - \ α and $ œ .α- $α Then $ is also a distance function. Thus, for any set \ and any distance function $ À \ ‚ \ Ä P, there is a set \ ª \ and $ ª $ with the property that if ?Ò$ Ð+ ” ,ÑÓ@, then ?Ò$ Ð+Ñ ” $ Ð,ÑÓ@, with witness sequence of length $ in general or length # when P is modular. It follows that $ preserves join in the restricted sense that $ Ð+ ” ,Ñ œ $ Ð+Ñ ” $ Ð,Ñ To finish the proof, note that if P does not have a smallest element, we can adjoin one to get a lattice P! . If P! is isomorphic to a sublattice of PartÐ\Ñ, then so is P. Therefore, we may as well assume that P has a smallest element. Define a distance function $! À P ‚ P Ä P by $! Ð+ß ,Ñ œ œ +”, ! if + Á , if + œ , and let \! œ P. Define a sequence of pairs Ð\5 ß $5 Ñ for 5 œ !ß "ß á by \5" œ \5 and then set and and $5" œ $5 \ œ .Ö\5 ± 5 œ !ß "ß á × $ œ .Ö$5 ± 5 œ !ß "ß á × Then $À \ ‚ \ Ä P is a distance function. Moreover, if Å is the $-extension and ?$ Ð+ ” ,Ñ@ for ?ß @ − \ , then ?ß @ − \5 for some 5 and so ?$5 Ð+ ” ,Ñ@, which implies that ?$5 Ð+ÑD" $5 Ð,ÑD# $ Ð+ÑD$ $ Ð,Ñ@ for some D" ß D# ß D$ − \ . Since $5 œ $5" © $ , we have ?$ Ð+ÑD" $ Ð,ÑD# $ Ð+ÑD$ $ Ð,Ñ@ and so ?Ò$ Ð+Ñ ” $ Ð,ÑÓ@. Hence, the embedding $ À P Ä PartÐ\Ñ is a lattice Modular and Distributive Lattices 119 embedding of type $. A similar argument holds for the #-extension when P is modular. Theorem 4.21 prompts us to wonder about type " representations. Definition Let B! ß B" ß B# ß C! ß C" ß C# be variables and let A3ß4 œ ÐB3 ” B4 Ñ • ÐC3 ” C4 Ñ A œ A!ß" • ÐA!ß# ” A"ß# Ñ The Arguesian identity is ÐB! ” C! Ñ • ÐB" ” C" Ñ • ÐB# ” C# Ñ Ÿ ÐÐA ” B" Ñ • B! Ñ ” ÐÐA ” C"Ñ • C!Ñ and a lattice that satisfies the Arguesian identity is Arguesian. It can be shown that the following implications hold, but their converses do not hold: Distributive Ê Arguesian Ê Modular Theorem 4.23 (Jónsson [37]) Any lattice that has a type " representation is Arguesian. Proof. We may assume that P is a sublattice of PartÐ\Ñ. Let +3 ß ,3 − P for 3 œ !ß "ß #. If Bß C − \ are related under Ð+! ” ,! Ñ • Ð+" ” ," Ñ • Ð+# ” ,# Ñ then BÐ+3 ” ,3 ÑC and so there exist ?! , ?" and ?# in \ for which B+3 ?3 ,3 C for 3 œ !ß "ß #. Let -3ß4 œ Ð+3 ” +4 Ñ • Ð,3 ” ,4 Ñ for ! Ÿ 3 4 Ÿ $. Then ?3 -3ß4 ?4 for all ! Ÿ 3 4 Ÿ $. Set - œ -!ß" • Ð-!ß# ” -"ß# Ñ Then ?! -?" and so BÒÐÐ- ” +" Ñ • +! ÑÓ?! ?! ÒÐÐ- ” ," Ñ • ,! ÑÓC and so B and C are related under the join 120 Lattices and Ordered Sets ÐÐ- ” +" Ñ • +! Ñ ” ÐÐ- ” ," Ñ • ,! Ñ as required by the Arguesian identity. The converse of Theorem 4.23 is not true. M. Haiman [29] and [30] has shown that there are Arguesian lattices that do not have type " representations. Note that the embedding AÀ P ä PartÐ\Ñ of Theorem 4.21 involves an infinite set \ and for many years, it remained an open question as to whether or not a finite lattice could be embedded into a finite partition lattice. In 1980, Pudlák º and Tuma [50] showed that this is the case, but the proof is quite difficult. One of the most interesting consequences of Theorem 4.23 is the following. Theorem 4.24 (Whitman [67]) Every lattice P is isomorphic to a sublattice of the lattice fÐKÑ of all subgroups of some group K. Proof. We may assume that P is a sublattice of PartÐ\Ñ. Let W\ be the group of all permutations of \ . For each + − P, let W+ be the subgroup of W\ generated by all transpositions ÐB CÑ for which B+C. If 7 − W+ , then 7 is a product of transpositions 7 œ ÐB" C" ÑâÐB8 C8 Ñ where B3 +C3 . If the transpositions on the right are not disjoint, then we may reorder (if necessary) and combine two adjacent transpositions with a common element. If the transpositons are identical, they cancel. If they have exactly one element in common, then we get a $-cycle in which all three elements are related under +. This process continues until we reach a product of disjoint cycles in which all elements in each cycle are related under +. In other words, the unique (up to order) cyclic decomposition of 7 has the property that all elements in a given cycle are related under +. In particular, any transposition Ð? @Ñ in W+ has the property that ?+@ and so a transposition Ð? @Ñ is in W+ if and only if ?+@, which implies that the map + È W+ is injective. Also, 7 − W+ ∩ W, if and only if the cycles in its unique cycle decomposition have the property that every element in each cycle is related by + and by , , and so by + • , . Thus, W+ ∩ W, œ W+•, . Finally, W+ © W+”, and W, © W+”, imply that W+ ” W, © W+”, . For the reverse inclusion, if 7 − W+”, , then 7 œ ÐB" C" ÑâÐB8 C8 Ñ where B3 +C3 or B3 ,C3 . Thus, each transposition ÐB3 C3 Ñ is in W+ or W, and therefore in W+ ” W, . Hence, 7 − W+ ” W, , which shows that W+ ” W, œ W+”, and so the map + È W+ is a lattice embedding from P into f ÐKÑ. Modular and Distributive Lattices 121 Distributive Lattices Let us explore some of the basic properties of distributive lattices. First, in a distributive lattice, maximal ideals are prime. Theorem 4.25 In a distributive lattice P, all maximal ideals are prime. Proof. Let Q be a maximal ideal of P. Suppose that + • , − Q but +  Q . Then Æ + ” Q œ P and so , Ÿ + ” 7 for some 7 − Q . Hence, , œ , • Ð+ ” 7Ñ œ Ð, • +Ñ ” Ð, • 7Ñ − Q which shows that Q is prime. In a distributive lattice, elements are join-irreducible if and only if they are joinprime. Theorem 4.26 Let P be a distributive lattice. 1) The following are equivalent for B − P: a) B is join-irreducible b) B is join-prime c) Å B is a prime filter, or equivalently, Ð Å BÑ- is a prime ideal. 2) Dually, the following are equivalent for B − P: a) B is meet-irreducible b) B is meet-prime c) Æ B is a prime ideal. Proof. If B is join-irreducible, then BŸ+”, Í B • Ð+ ” ,Ñ œ B Í ÐB • +Ñ ” ÐB • ,Ñ œ B and so B ” + œ B or B • , œ B, that is, B Ÿ + or B Ÿ , . Hence, B is join-prime. Conversely, if B is join-prime and B œ + ” , , then B Ÿ + ” , and so we may assume that B Ÿ +, in which case B Ÿ + Ÿ B, that is, B œ +. Hence, B is joinirreducible. The rest follows from Theorem 3.37. We have seen that the down map 9Æ is an order embedding and so its restriction 9Æ À `ÐPÑ ä SpecÐPÑ is an order embedding of meet-irreducibles into prime ideals. Similarly, the up map is an anti-embedding and so the not-up map 9cÅ À ] ÐPÑ Ä SpecÐPÑ defined by 9cÅ Ð+Ñ œ Ð Å +Ñis an order embedding of join-irreducibles into prime ideals. More can be said if P has a chain condition. Theorem 4.27 Let P be a distributive lattice. 1) The not-up map 9cÅ À ] ÐPÑ Ä SpecÐPÑ defined by 9cÅ Ð+Ñ œ Ð Å +Ñ- 122 Lattices and Ordered Sets is an order embedding. If P has the DCC, then all prime ideals have the form Ð Å +Ñ- for + − ] ÐPÑ and so 9cÅ À ] ÐPÑ ¸ SpecÐPÑ is an order isomorphism. 2) The down map 9Æ À `ÐPÑ Ä SpecÐPÑ defined by 9Æ Ð7Ñ œ Æ 7 is an order embedding. If P has the ACC, then 9Æ À `ÐPÑ ¸ SpecÐPÑ is an order isomorphism. 3) If P is finite, then 9Æ ‰ 9cÅ À ] ÐPÑ ¸ `ÐPÑ and so ] ÐPÑ and `ÐPÑ are isomorphic. Proof. For part 1), if P has the DCC, then the prime filters are principal and so have the form Å +, where + is join-irreducible. Hence, the prime ideals (being complements of the prime filters) have the form Ð Å +Ñ- , where + is joinirreducible. Thus, 9cÅ is surjective. Part 2) is similar. Distributive lattices also have nice properties with respect to finiteness. Theorem 4.28 For a distributive lattice P, the following are equivalent: 1) P is finite 2) P has finite length 3) P has no infinite chains (equivalently, P has both chain conditions). Proof. Clearly a finite lattice has finite length and a lattice with finite length has no infinite chains. Suppose that P has no infinite chains. Since P has the DCC, Theorem 3.29 implies that the set ] ÐPÑ of join-irreducibles in P is finite-joindense in P. Hence, it is sufficient to show that ] ÐPÑ is a finite set. Certainly, ] ÐPÑ has no infinite chains and so Theorem 1.14 implies that it is sufficient to show that ] ÐPÑ has no infinite antichains. If E œ Ð+" ß +# ß á Ñ is an infinite antichain in ] ÐPÑ, the ACC on P implies that the sequence of joins +" ß + " ” + # ß + " ” + # ” + $ ß á is eventually constant and so there is an 8 ! for which +" ” â ” +8 ” +8" œ +" ” â ” +8 that is, +8" Ÿ +" ” â ” +8 Since +8" is join-irreducible, it is join-prime and so +8" Ÿ +5 for some 5 Ÿ 8, Modular and Distributive Lattices 123 contradicting the fact that E is an antichain. Hence, ] ÐPÑ has no infinite antichains. Irredundant Join-Irreducible Representations Theorem 3.29 implies that if a lattice P has the DCC, then ] ÐPÑ is finite-joindense in P. For + − P, an expression of the form + œ ?" ” â ” ? 8 where the ?3 's are distinct join-irreducibles is irredundant if + is not the join of any proper subset of Y œ Ö?" ß á ß ?8 ×. Clearly, if the join is irredundant, then Y must be an antichain. Moreover, if P is distributive, then the join is irredundant if and only if Y is an antichain. Let us refer to such a representation of + as an irredundant join-irreducible representation of +. In the presence of modularity, we can say some things about the uniqueness of irredundant join-irreducible representations of elements. Theorem 4.29 (Kurosh–Ore theorem, [40], [48]) Let P be a modular lattice. Suppose that + − P has two irredundant join-irreducible representations + œ B" ” â ” B 8 œ C" ” â ” C 7 1) (Kurosh–Ore replacement property) For any B3 , we can substitute some C4 to get another join-irreducible representation of +. 2) 8 œ 7 Thus, if P has the DCC, then every element + − P has an irredundant joinirreducible representation + œ B" ” â ” B 8 and all such representatioms have the same number of terms. Proof. For part 1), it is sufficient to prove the theorem for B" . Let B s " œ B# ” â ” B 8 and -5 œ C5 ” B s" Then we must find an index 5 for which -5 œ +. Now, it is true that + œ -" ” â ” -7 However, + is join-irreducible in the interval ÒB s" ß +Ó, which contains each -5 , because the modularity of P implies that ÒB" • B s" ß B" Ó ¸ ÒB s" ß B" ” B s" Ó œ ÒB s" ß +Ó and because B" is join-irreducible in the interval ÒB" • B s" ß B" Ó. Thus, + œ -5 for some 5 . For part 2), suppose that 8 7. Then we can replace all of the B's by C's to get a shorter join-irreducible representation of + using only C's, which contradicts the irredundancy of the original representation of + in terms of the C's. 124 Lattices and Ordered Sets When P is distributive, there is uniqueness, since B" ” â ” B 8 œ C " ” â ” C 8 implies that C3 Ÿ B " ” â ” B 8 and so C3 Ÿ B4 for some 4. By a symmetric argument, C3 Ÿ B4 Ÿ C5 for some 5 and since ÖC" ß á ß C8 × is an antichain, equality must hold and so ÖB" ß á ß B8 × œ ÖC" ß á ß C8 × Theorem 4.30 Let P be a distributive lattice. 1) Every + − P has at most one irredundant join-irreducible representation + œ B" ” â ” B 8 2) If P has the DCC, then every + − P has exactly one irredundant joinirreducible representation. Exercises 1. 2. 3. Prove that the modular law is self-dual. Find an infinite lattice with no infinite chains. Show that the lattice of normal subgroups of the noncyclic four-group Z œ Ö"ß +ß ,ß +,× 4. is isomorphic to Q$ , which is not distributive. If K is a group, prove that f ÐKÑ need not be modular. Hint: Consider the alternating group E% of order "#. Let E œ ØÐ" #ÑÐ$ %ÑÙß F œ ØÐ" #ÑÐ$ %Ñß Ð" $ÑÐ# %ÑÙ and G œ ØÐ" # $ÑÙ 5. 6. Prove that the family a ÐKÑ of normal subgroups of a group K is modular. Prove that a lattice is distributive if and only if + ” Ð, • -Ñ Ð+ ” ,Ñ • - 7. for all +ß ,ß - − P. Let P be a distributive lattice and let E and F be ideals in P. Prove that E ” F œ Ö+ ” , ± + − Eß , − F× 8. Prove that a lattice is distributive if and only if the map 9+ À P Ä P ‚ P defined by 9+ ÐBÑ œ ÐB • +ß B ” +Ñ is a lattice embedding, for all + − P. Modular and Distributive Lattices 9. 125 Prove that a lattice is distributive if and only if for all +ß ,ß - − P, the selfdual median property holds Ð+ • ,Ñ ” Ð+ • -Ñ ” Ð, • -Ñ œ Ð+ ” ,Ñ • Ð+ ” -Ñ • Ð, ” -Ñ 10. If P is a distributive lattice, show that an expression + œ ?" ” â ” ? 8 with ?3 join-irreducible is irredundant if and only if the terms are distinct and form an antichain. 11. Prove that a finite lattice P is modular if and only if for all +ß , − P, the opposite sides of the quadrilateral in Figure 4.4 are isomorphic, that is, Ò+ • ,ß +Ó ¸ Ò,ß + ” ,Ó 12. Is a relatively complemented lattice complemented? 13. Show that the subspaces of a vector space Z form a complemented modular lattice under set inclusion. Describe the relative complements of elements. Is the lattice distributive? 14. Prove that the lattice Ðß ± Ñ is distributive. 15. Prove that the lattice of subgroups of a finite cyclic group K of order 8 is distributive. 16. Using the fact that if the lattice fÐKÑ is distributive then K is cyclic, describe all finite groups whose lattice of subgroups is isomorphic to a power set lattice kÐ\Ñ. 17. Let P be a bounded distributive lattice. Prove that the set W of complemented elements of P form a sublattice of P. 18. Let \ be a nonempty set. Let c be the family of all partitions of \ . Order c as follows: - Ÿ 5 if - is a refinement of 5 , that is, if every block of 5 is a union of blocks of -. a) Show that Ðc ß Ÿ Ñ is a complete lattice. Describe the meet and join. b) Describe the largest and smallest elements of c . c) Describe the covering relation. d) Show that the height of a partition - is equal to k\ k k-k. e) Prove that if 8 % then c is not modular. 19. Let P be a distributive lattice. If + and +w are complementary then + • +w œ !. Prove that +w is the largest B for which + • B œ !. 20. Let P be a distributive lattice in which ] ÐPÑ œ TÐPÑ and TÐPÑ is joindense. Suppose that + − P has a complement +w . Let TÆ Ð+Ñ œ ÖB − TÐPÑ ± B Ÿ +× and TÆ Ð+w Ñ œ ÖB − TÐPÑ ± B Ÿ +w × Prove that TÐPÑ œ TÆ Ð+Ñ ∪ TÆ Ð+w Ñ. 126 Lattices and Ordered Sets 21. Let P be a distributive lattice and let M be a proper ideal of P and let J be a proper filter in P. Suppose that Q is a maximal element of the set of all ideals of P that contain M and are disjoint from J . Prove that Q is prime in P. Hint: Suppose that + • , − Q but that +ß ,  Q and consider the ideals NB œ Æ ÖB ” 7 ± 7 − Q × for B œ + and B œ , . 22. Let P be a lattice and let E © P be a nonempty subset of P. Then the lattice generated by E, denoted by ØEÙ, is defined to be the intersection of all sublattices of P containing E. If P is distributive, show that ØEÙ can be constructed as follows: First, take the meets of all nonempty finite subsets of E. Then ØEÙ is the set of joins of all nonempty finite subsets of these meets. 23. Prove part 2) of Theorem 4.30 using duality. The Partition Lattice 24. Let K be a group. Show that there is a very well known representation of the lattice a ÐKÑ of normal subgroups of K in PartÐKÑ. Show that this representation is of type " and so a ÐKÑ is Arguesian. 25. Prove that PartÐ\Ñ is strongly atomic. 26. Prove that PartÐ\Ñ is semimodular. 27. Prove that PartÐ\Ñ is relatively complemented. 28. Prove that an Arguesian lattice is modular by showing that R& is not Arguesian. Hint: let B! œ C" œ C B" œ B # œ C ! œ D C# œ B Pseudocomplements Let P be a lattice with a smallest element. For + − P, an element B − P such that +•Bœ! is called a semicomplement of +. The set f Ð+Ñ of semicomplements of + is clearly a down-set. If f Ð+Ñ is a principal ideal, that is, if f Ð+Ñ has a greatest element, this element is called a pseudocomplement of +. A lattice in which every element has a pseudocomplement is called a pseudocomplemented lattice. More generally, if +ß , − P and if the set ÖB − P ± + • B Ÿ ,× has a greatest element, it is called a relative pseudocomplement of + in , . We Modular and Distributive Lattices 127 denote this element by + , . (There does not appear to be a generally accepted notation for relative pseudocomplement.) If every pair of elements has a relative pseudocomplement, then P is said to be relatively pseudocomplemented. A relatively pseudocomplemented lattice is also called a Heyting lattice or a Heyting algebra. The dual of a Heyting algebra is a lattice P that satisfies the following condition: For every +ß , − P, the set ÖB − P ± + ” B ,× has a smallest element, which we will denote by + , . If + , exists for every pair of elements in P, then P is called a Brouwerian lattice. (Birkhoff refers to Heyting lattices as Brouwerian lattices but others refer to the dual of a Heyting algebra as a Brouwerian lattice.) 29. Prove that every Brouwerian or Heyting lattice is distributive. 30. Show that any finite distributive lattice is Brouwerian and Heyting. 31. Let P be a distributive lattice. Suppose that + and +w are complementary in P. a) Prove that + , œ +w ” , . b) Show that the ordinary complement +w is a special type of relative pseudocomplement. 32. Let P œ Ðß ± Ñ. Show that + , exists for all +ß , − P but that the relative pseudocomplement + , does not exist. Thus, P is Brouwerian but not Heyting. 33. Let T be a finite poset. Let P œ bÐT Ñ be the lattice of down-sets in T . Show that P is pseudocomplemented. Hint: What does it imply to say that a down-set H is disjoint from a down-set E? Chapter 5 Boolean Algebras Boolean Lattices Complemented distributive lattices have particularly nice properties. Definition A complemented distributive lattice is called a Boolean lattice. Since a distributive lattice has the property that the sets +Ò?ß@Ó are empty or singletons, Boolean lattices are uniquely complemented. Some simple properties of Boolean lattices follow. Theorem 5.1 Let P be a Boolean lattice. Let +ß , − P. Then 1) !w œ "ß "w œ ! and +ww œ + 2) (de Morgan's laws) Ð+ • ,Ñw œ +w ” , w Ð+ ” ,Ñw œ +w • , w 3) (Order and complements) +Ÿ, Í + • ,w œ ! Í +w ” , œ " In particular, 3a) (Order-reversing) +Ÿ, Í , w Ÿ +w 3b) (Atoms) If + − TÐPÑ, then for any B − P, +ŸB or + Ÿ Bw but not both Proof. For part 3), if + Ÿ , , then + • , w is less than or equal to both , and , w and so + • , w œ !. Conversely, if + • , w œ !, then S. Roman (ed.), Lattices and Ordered Sets, doi: 10.1007/978-0-387-78901-9_5 , © Steven Roman 2008 129 130 Lattices and Ordered Sets , œ , ” Ð+ • , w Ñ œ Ð, ” +Ñ • Ð, ” , w Ñ œ , ” + and so + Ÿ , . The other part is dual. We leave the rest of the proof to the reader. Theorem 3.45 (projecting a complement into an interval to get a relative complement) implies that a Boolean lattice is relatively complemented. Therefore, distributivity implies that P is uniquely relatively complemented. Theorem 5.2 A Boolean lattice P is uniquely relatively complemented. Hence, the following hold in P: 1) For any + − Ò?ß @Ó, we have +Ò?ß@Ó œ Ð+w ” ?Ñ • @ œ Ð+w • @Ñ ” ? 2) The join-irreducible elements of P are precisely the atoms of P, that is, ] ÐPÑ œ TÐPÑ 3) A Boolean lattice is atomic if and only if it is atomistic. We have seen that in a distributive lattice, all maximal ideals are prime. In a Boolean lattice, the converse is also true. Theorem 5.3 Let P be a Boolean lattice and let M be a proper ideal of P. The following are equivalent: 1) M is maximal. 2) M is prime. 3) For all + − P, exactly one of + or +w is in M . Proof. We have seen that maximal ideals are prime. If M is prime, then + • +w − M and so + − M or +w − M . However, if both are in M , then so is + ” +w œ ", which is false since M is proper. Finally, if 3) holds and M § N , then for + − N Ï M , we have +w − M and so " œ + ” +w − N , whence N œ P. This shows that M is maximal. Boolean Algebras Just as a lattice can be defined algebraically as a nonempty set with two operations • and ” satisfying certain requirements, with no explicit mention of the underlying partial order, Boolean lattices can also be defined algebraically as a set with operations. In this context, the term Boolean algebra is usual. Definition A Boolean algebra ÐFß • ß ” ß !ß "ß w Ñ is a sextuple, where F is a nonempty set with two binary operations • and ” , one unary operation w and two nullary operations (constants) ! and ", having the following properties: Boolean Algebras 131 1) • and ” are associative, commutative, idempotent and satisfy the absorption laws, that is, the triple ÐFß • ß ” Ñ is a lattice as defined algebraically in Theorem 3.17. 2) • and ” satisfy the distributive laws. 3) ! is the identity for join and " is the identity for meet, that is, +”!œ+ and + • " œ + for all + − P. 4) The unary operation w satisfies the definition of complement: + ” +w œ " and + • +w œ ! Thus, a Boolean algebra ÐFß • ß ” ß w ß !ß "Ñ is a Boolean lattice with smallest element ! and largest element ". Conversely, any Boolean lattice is a Boolean algebra. Definition A nonempty subset W of a Boolean algebra F is a Boolean subalgebra if W inherits the bounds ! and " as well as finite meets, joins and complements from F . Example 5.4 1) The power set kÐWÑ of a set W is a Boolean algebra, called the power set algebra, or just the power set of W . In fact, these are very general algebras: We will prove in a later chapter that every finite Boolean algebra is isomorphic to kÐWÑ for some finite set W and, more generally, every complete atomic Boolean algebra is isomorphic to a power set kÐWÑ. A Boolean subalgebra F of a power set algbera kÐWÑ is called a power set subalgebra or an algebra of sets or a field of sets. Put another way, a family Y of sets is an algebra of sets if it has finite unions, finite intersections and complements. 2) Let W be a nonempty set. The finite-cofinite algebra J GÐWÑ is the family of all subsets of W that are either finite or cofinite. (A subset is cofinite if its complement is finite.) For a countably infinite set W , this Boolean algebra is not isomorphic to any power set kÐ\Ñ because kJ GÐWÑk œ i! whereas kkÐ\Ñk is either finite or uncountable. Alternatively, we can observe that J GÐWÑ is not complete for any infinite set W , but any lattice of the form kÐ\Ñ is complete. Boolean Rings There is yet another way to view Boolean algebras algebraically. Definition A Boolean ring is a ring ÐVß ß † ß !ß "Ñ with identity " in which every element is idempotent, that is, <# œ < for all < − V . 132 Lattices and Ordered Sets We leave proof of the following to the reader. Theorem 5.5 A Boolean ring is commutative and #< œ ! for all < − V . Theorem 5.6 1) Let ÐVß ß † ß !ß "Ñ be a Boolean ring. Define binary operations • and ” and a unary operation w in V by < • = œ <= < ” = œ < = <= <w œ " < Then ÐVß • ß ” ß !ß "ß w Ñ is a Boolean algebra. 2) Conversely, let ÐFß • ß ” ß !ß "ß w Ñ be a Boolean algebra. Define binary operations by <= œ < • = < = œ Ð< ” =Ñ • Ð<w ” =w Ñ œ Ð< • =w Ñ ” Ð<w • =Ñ Then ÐFß ß † ß !ß "Ñ is a Boolean ring. The maps αÀ ÐVß ß † ß !ß "Ñ È ÐVß • ß ” ß !ß "ß w Ñ and 3À ÐVß • ß ” ß !ß "ß w Ñ È ÐVß ß † ß !ß "Ñ described above are inverses of each other. Moreover, a nonempty subset W © V is a lattice ideal if and only if it is a ring ideal. Proof. For part 1), it is clear that meet is associative, commutative and idempotent and that join is commutative and idempotent. Verification of associativity of join is straightforward, and is left to the reader. Finally, the absorption laws hold since + • Ð+ ” ,Ñ œ +Ð+ , +,Ñ œ + + ” Ð+ • ,Ñ œ + +, ++, œ + Note that the absorption laws do not hold if join is defined simply as addition. For part 2), let F be a Boolean algebra. As to addition, a straightforward but tedious calculation shows that addition is associative. The zero of F is !, since < ! œ Ð< ” !Ñ • Ð<w ” "Ñ œ < It is clear that addition is commutative and so F is an abelian group. It is clear that multiplication is associative and the identity of F is ". We leave verification of distributivity Boolean Algebras 133 <Ð= >Ñ œ <= <> for the reader. Finally, <# œ < • < œ < and so F is a Boolean ring. Given a Boolean algebra U œ ÐFß • ß ” ß !ß "ß w Ñ, 3ÐU Ñ œ ÐFß ß † ß !ß "Ñ satisfies the Boolean ring <= œ < • = < = œ Ð< ” =Ñ • Ð<w ” =w Ñ In the Boolean algebra αÐ3ÐU ÑÑ œ ÐFß • ß ” ß !ß "ß w Ñ, we have <w œ " < œ Ð" ” <Ñ • Ð! ” <w Ñ œ <w and so complements in U and αÐ3ÐU ÑÑ are the same. Meets are also the same, since < • = œ <= œ < • = For the join, first note that Ð< =Ñw œ ÒÐ< ” =Ñ • Ð<w ” =w ÑÓw œ Ð< • =Ñ ” Ð<w • =w Ñ and so < ” = œ < = <= œ ÒÐ< ” =Ñ • Ð<w ” =w ÑÓ Ð< • =Ñ œ ÖÒÐ< ” =Ñ • Ð<w ” =w ÑÓ ” Ð< • =Ñ× • ÖÒÐ< • =Ñ ” Ð<w • =w ÑÓ ” Ð<w ” =w Ñ× œ Ð< ” =Ñ • ÒÐ<w ” =w Ñ ” Ð< • =ÑÓ • ÒÐ< • =Ñ ” Ð<w ” =w ÑÓ œ Ð< ” =Ñ • Ò<w ” =w ” Ð< • =ÑÓ œ Ð< ” =Ñ • ÒÐ< • =Ñw ” Ð< • =ÑÓ œ<”= Thus, meet, join and complement are the same in U and αÐ3ÐU ÑÑ, which implies that the order is the same and so ! œ ! and " œ ". Thus, αÐ3ÐU ÑÑ œ U as Boolean algebras and so α ‰ 3 œ +. Given a Boolean ring e œ ÐVß ß † ß !ß "Ñ, the Boolean algebra αÐeÑ satisfies < • = œ <= < ” = œ < = <= <w œ " < In the corresponding Boolean ring 3Ð5ÐeÑÑ œ ÐFß ß † ß !ß "Ñ, we have 134 Lattices and Ordered Sets < † = œ < • = œ <= and < = œ Ð< • =w Ñ ” Ð<w • =Ñ œ <=w ” <w = œ <=w <w = <=w <w = œ <Ð" =Ñ Ð" <Ñ= <Ð" =ÑÐ" <Ñ= œ < <= = <= <=Ð" = < <=Ñ œ < <= = <= <= <= <= <= œ<= Since addition and multiplication are the same in e and αÐ3ÐeÑÑ, we have e œ αÐ3ÐeÑÑ as rings and so 3 ‰ α œ +. Finally, let ÐVß ß † ß !ß "Ñ be a Boolean ring and let ÐVß • ß ” ß !ß "Ñ be the corresponding Boolean algebra. To see that a ring ideal M © V is a lattice ideal, let +ß , − M . Then + ” , œ + , +, − M and if < Ÿ +, then < œ < • + œ <+ − M and so M is a down-set. Thus, M is a lattice ideal. Conversely, if M © V is a lattice ideal, then for +ß , − M , + , œ Ð+ • , w Ñ ” Ð+w • ,Ñ − M Also, if < − V and + − M , then <+ œ < • + − M Hence, M is a ring ideal. Boolean Homomorphisms Boolean homomorphisms preserve the Boolean algebra structure. Definition Let P and Q be Boolean algebras. A Boolean homomorphism from P to Q is a function 0 À P Ä Q with the following properties: 1) 0 preserves meets and joins, that is, 0 Ð+ • ,Ñ œ 0 Ð+Ñ • 0 Ð,Ñ 0 Ð+ ” ,Ñ œ 0 Ð+Ñ ” 0 Ð,Ñ 2) 0 preserves ! and ", that is, 0! œ ! and 0" œ " 3) 0 preserves complements, that is, 0 Ð+w Ñ œ Ð0 +Ñw Boolean Algebras 135 It is not hard to see that in a uniquely complemented lattice, preservation of bounds (! and ") is equivalent to preservation of complements. Theorem 5.7 The following are equivalent for a lattice homomorphism 0 À P Ä Q between uniquely complemented lattices: 1) 0 preserves complements, that is, 0 Ð+w Ñ œ Ð0 +Ñw for all + − P. 2) 0 is a Ö!ß "×-homomorphism, that is, 0 Ð!Ñ œ ! and 0 Ð"Ñ œ " Proof. This follows easily from the equations 0 Ð!Ñ œ 0 Ð+ • +w Ñ œ 0 Ð+Ñ • 0 Ð+w Ñ 0 Ð"Ñ œ 0 Ð+ ” +w Ñ œ 0 Ð+Ñ ” 0 Ð+w Ñ Note that a Boolean homomorphism is just a lattice Ö!ß "×-homomorphism between Boolean algebras. One place to find Boolean homomorphisms is in induced maps. If 0 À E Ä F is a function between nonempty sets, the induced inverse map 0 " À kÐFÑ Ä kÐEÑ is defined by 0 " ÐZ Ñ œ Ö+ − E ± 0 Ð+Ñ − Z × If 0 is bijective, the symbol 0 " usually denotes the ordinary inverse map defined on the elements of F , rather than the induced inverse map defined on the power set kÐFÑ. We leave proof of the following to the reader. Theorem 5.8 Let 0 À E Ä F be a function between nonempty sets. The induced inverse map 0 " À kÐFÑ Ä kÐEÑ is a Boolean homomorphism. In fact, for any subset Y of F and any family ÖY3 ± 3 − M× of subsets of F , we have 0 " Ð.Y3 Ñ œ .0 " ÐY3 Ñ 0 " Ð,Y3 Ñ œ ,0 " ÐY3 Ñ 0 " ÐY w Ñ œ Ò0 " ÐY ÑÓw Note that if F is a Boolean lattice, then its dual F is also a Boolean lattice, with meet in F ` equal to join in F and join in F ` equal to meet in F . However, complement is self-dual: The complement in F ` is the same as the complement in F . ` 136 Lattices and Ordered Sets In a Boolean lattice F , the complement map 5À B È Bw is order-reversing and bijective and so is a lattice-isomorphism from F to its dual F ` . Also, de Morgan's laws say precisely that 5 is a Boolean algebra isomorphism. Theorem 5.9 Let F be a Boolean lattice. The complement map B È Bw is a Boolean isomorphism from F onto the dual F ` . Characterizing Boolean Lattices We have discussed the fact that if a lattice P is uniquely relatively complemented, then it is distributive, but that being uniquely complemented it not sufficient to imply distributivity. This follows from a theorem of Robert Dilworth. Theorem 5.10 (Dilworth, 1945 [16]) Every lattice is a sublattice of a uniquely complemented lattice. Thus, we are prompted to look at conditions that imply that a uniquely complemented lattice is distributive, and therefore Boolean. One such condition is the following special form of modularity. Definition Let P be a complemented lattice. 1) P satisfies SMP1 (for special modular property) if + - Ê + • Ð+w ” -Ñ œ - Ê + ” Ð+w • -Ñ œ - for all +ß - − P. 2) Dually, P satisfies SMP2 if - + If P satisfies both SMP1 and SMP2, we say that P satisfies SMP. We can now describe conditions that characterize distributivity (and therefore Booleanness) among uniquely complemented lattices. Theorem 5.11 Let P be a uniquely complemented lattice. The following are equivalent: 1) P is distributive (and therefore a Boolean lattice). 2) P is modular (due to Birkhoff and von Neumann). 3) P satisfies SMP. 4) P satisfies de Morgan's laws. Proof. It is clear that 1) Ê 2) Ê 3). To see that 3) Ê 1), we show that the sets +Ò?ß@Ó are empty or singletons, which implies distributivity by Theorem 4.8. Actually, SMP is precisely what is needed to prove that any relative complement is also a complement and is therefore unique by assumption. Boolean Algebras 137 Specifically, we will show that if , − +Ò?ß@Ó , then , is the complement of the element / œ Ð+ • ?w Ñ ” @w which is formed by projecting + into the interval Ò@w ß ?w Ó. First, SMP1 and the fact that @ + • ?w imply that / • , Ÿ / • @ œ ÒÐ+ • ?w Ñ ” @w Ó • @ œ + • ?w and so / • , is contained in +ß , and ?w and since + • , œ ?, it follows that / • , is contained in ? and ?w , whence / • , œ !. Dually, SMP2 implies that / ” , / ” ? œ @w ” Ð+ • ?w Ñ ” ? œ @w ” + Hence, / ” , contains @w ß + and , and since + ” , œ @, it follows that / ” , contains @ as well, and so / ” , œ ". Thus, , œ /w and 1)–3) are equivalent. We leave it to the reader to show that the distributive law implies de Morgan's laws. Finally, we show that de Morgan's laws imply SMP. For SMP2, it is sufficient to show that B + Ê ÒÐB • +w Ñ ” +Ów œ Bw which, using de Morgan's laws, is B + Ê ÐB • +w Ñw • +w œ Bw The consequent is equivalent to B • ÒÐB • +w Ñw • +w Ó œ ! B ” ÒÐB • +w Ñw • +w Ó œ " The first of these equations is true. For the second equation, since Bw Ÿ +w , we have B ” ÒÐB • +w Ñw • +w Ó œ B ” ÒÐBw ” +Ñ • +w Ó B ” ÒÐBw ” +Ñ • Bw Ó œ B ” Bw œ " Hence, SMP2 holds. Condition SMP1 follows by duality. We also have the following result. Theorem 5.12 1) (Birkhoff and Ward [9]) A uniquely complemented atomic lattice is distributive. 2) A uniquely complemented lattice of finite width is distributive. Proof. For part 1), let P be a uniquely complemented lattice and let T œ TÐPÑ be the set of atoms of P. We first show that the T-down map 9TßÆ À P Ä kÐPÑ preserves strict order, that is, 138 Lattices and Ordered Sets BC Ê TÆ ÐBÑ § TÆ ÐCÑ It then follows from Theorem 3.32 that the T-down map is injective. Now, if B C and Bw is the complement of B, then Bw ” C œ ". It follows that Bw • C !, since otherwise, C and B would be distinct complements of Bw . Hence, there is an atom + Ÿ Bw • C. In particular, + Ÿ C and + Ÿ Bw . Hence, + Ÿ y B and so + − TÆ ÐCÑ Ï TÆ ÐBÑ. Thus, the T-down map is injective. The T-down map is a meet-homomorphism (as are all F -down maps). To show that the T-down map is a join-homomorphism, we show that TÆ ÐB ” CÑ © TÆ ÐBÑ ∪ TÆ ÐCÑ since the reverse inclusion is clear. This is equivalent to +ŸB”C Ê +ŸB or + Ÿ C for all atoms +. If + is not contained in B, then Theorem 3.43, which says that every atom is contained in the complement of all atoms other than itself, implies that the atoms contained in B are also contained in +w and so TÆ ÐBÑ œ TÆ Ð+w • BÑ which implies that B œ +w • B, that is, B Ÿ +w . Similarly, if + Ÿ y C, then C Ÿ +w . w But then + Ÿ B ” C Ÿ + , which is false. Hence, either + Ÿ B or + Ÿ C . Thus, the T-down map is a lattice embedding of P into the distributive lattice kÐPÑ and so P is distributive (in fact, P is isomorphic to a power set sublattice). To prove part 2), we show that P has the DCC, in which case it is atomic and so part 1) completes the proof. If P does not have the DCC, then P has a strictly descending sequence +" + # â w w w If ,5 œ +5 • +5" , then +5 ” +5" +5" ” +5" œ " and so ,5 cannot be equal w w to !, since otherwise, we have +5 œ +5" , which implies that +5 œ +5" . Thus, ,5 Á ! for all 5 . Moreover, for 5 4, w w ,5 • ,4 Ÿ ,5 • ,5" œ Ð+5 • +5" Ñ • Ð+5" • +5# Ñœ! and so the infinite family Ö,5 × is an antichain, which is not possible. Thus, P has the DCC. Complete and Infinite Distributivity We now wish to look at various generalizations of distributivity to infinite meets or joins. Boolean Algebras 139 Infinite Distributivity Perhaps the simplest generalization is given by the following definition. Definition Let P be a complete lattice. 1) P is join-infinitely distributive if for any B − P and any Ö+3 ± 3 − M× © P, B• 2 +3 3−M 2) P is meet-infinitely Ö+3 ± 3 − M× © P, œ 2 B • +3 3−M distributive B” 4 +3 3−M if for any B−P and any œ 4 B ” +3 3−M We say that P is infinitely distributive if both 2) and 3) hold. We leave it as an exercise to show that the join-infinite distributive law implies that 2 +3 • 3−M 2 ,4 4−N œ 2 Ð+3 • ,4 Ñ Ð3ß4Ñ−M‚N and similarly for meets. Example 4.5 shows that if E is the set of odd natural numbers in the complete lattice Ðß ± Ñ, then #• 2+ +−E Á 2 Ð# • +Ñ +−E and so a complete distributive lattice need not be infinitely distributive. However, the infinite distributive laws hold in a Boolean lattice, whenever the relevant meets and joins exist. Theorem 5.13 In a Boolean algebra, the infinite meet and join distributive laws hold whenever the relevant meets and joins exist. In particular, a complete Boolean algebra is infinitely distributive. Proof. Let F be a Boolean algebra and let Ö+3 ± 3 − M× © F . To establish joininfinite distributivity, it is clear that for any B − F , 2ÐB • +3 Ñ Ÿ B • Š2+3 ‹ For the reverse inequality, note that in any Boolean algebra we have 140 Lattices and Ordered Sets + Ÿ Bw ” + œ Bw ” Ð+ • BÑ and so 2+3 Ÿ 2ÐBw ” Ð+3 • BÑÑ Ÿ Bw ” Š2ÐB • +3 Ñ‹ Hence, (finite) distributivity gives B • Š2+3 ‹ Ÿ B • ’Bw ” Š2ÐB • +3 Ñ‹“ œ B • Š2ÐB • +3 Ñ‹ Ÿ 2ÐB • +3 Ñ A dual argument will establish meet-infinite distributivity. Complete Distributivity Consider now an arbitrary meet of arbitrary joins in a lattice P. Let us index the joins by an index set X and so, for each > − X , we have B> œ 2 +>ß3 3−M> where +>ß3 − P. Thus, the full expression is B œ 4 2 +>ß3 >−X 3−M> The most general form of distributivity can be described in words as follows: Choose a term from each join B> , take the meet of these terms and then take the join of all such meets. Moreover, the choice can be accomplished by a choice function, which can be thought of as a function 0 that maps each > − X to an element of the index set M> , thus choosing the member B>ß0 Ð>Ñ from B> . The maps 0 À X Ä -M> for which 0 Ð>Ñ − M> are precisely the elements of the cartesian product #M> . Hence, this form of distributivity can be expressed as follows: 4 2 +>ß3 >−X 3−M> œ 2 0 −#M> 4 +>ß0 Ð>Ñ >−X Definition Let P be a complete lattice. Let ÖM> ± > − X × be any family of nonempty sets and let +>ß3 − P for each > − X and 3 − M> . Then P is completely distributive if the following hold: Boolean Algebras 4 2 +>ß3 œ 2 2 4 +>ß3 œ 4 >−X 3−M> 0 −#M> 141 4 +>ß0 Ð>Ñ (5.14) 2 +>ß0 Ð>Ñ (5.15) >−X and, dually, >−X 3−M> 0 −#M> >−X It is not hard to see that (5.14) implies join-infinite distributivity and (5.15) implies meet-infinite distributivity. Let F be a complete Boolean algebra and assume that (5.14) holds. Consider the special case of the meet of the joins , ” , w , for all , − F : 4 Ð, ” , w Ñ ,−F Of course, this is equal to ". If we index the set Ö,ß , w × by setting , œ ,! and , w œ ," , then M, œ Ö!ß "× for all , − F and so we do not need double indexing on the elements of F . Complete distributivity gives " œ 4 Ð, ” , w Ñ œ 2 ,−F Each term 0 −#M, 4 ,0 Ð,Ñ ,−F B0 œ 4 ,0 Ð,Ñ ,−F is the meet of the set of elements described as follows: For each , − F , choose , or , w . If B0 Á !, then it must be an atom, for if , B0 , then none of the terms in B0 is equal to , and so one of the terms is , w , whence , B0 Ÿ , w and so , œ !. Thus, B0 œ ! or B0 is an atom. It follows that " is the join of atoms, say " œ 2 +3 Then by join-infinite distributivity, for any nonzero , − F , we have , œ , • Š2+3 ‹ œ 2Ð, • +3 Ñ and so , is the join of atoms. Thus, F is atomistic (and atomic). The converse is also true. Indeed, we will see in a later chapter that a complete atomic Boolean algebra is isomorphic to a power set algebra, which is completely distributive. 142 Lattices and Ordered Sets Theorem 5.16 Let F be a complete Boolean algebra. 1) F satisfies (5.14) if and only if F is atomic. 2) F is completely distributive if and only if it is atomic. Exercises 1. 2. 3. 4. 5. 6. 7. 8. 9. Prove that for an infinite set \ , the finite-cofinite algebra J GÐ\Ñ is not complete. Prove that if F is a Boolean lattice then its dual F ` is also a Boolean lattice, with meet in F ` equal to join in F and join in F ` equal to meet in F . Prove that complement is self-dual: the complement in F ` is the same as the complement in F . If F is a Boolean algebra show that + − F is an atom if and only if +w is a coatom. Let F be a Boolean lattice. If E is a nonempty subset of F , describe the Boolean sublattice ØEÙ generated by E, that is, the smallest Boolean sublattice of F containing E. (Thus, ØEÙ contains all finite joins, meets and complements formed in F using elements of E.) Prove that a Boolean lattice with the DCC also has the ACC. Prove that a Boolean algebra F is finite if and only if " is the join of a finite number of atoms. Prove that a Boolean ring is commutative and that #< œ ! for all < − V . In the correspondence between ring ideals of a Boolean ring V and lattice ideals in the Boolean lattice V , what can be said about the properties of being principal, prime or maximal? Those who enjoy arithmetic with meets and joins can show that the operation < = œ Ð< • =w Ñ ” Ð<w • =Ñ in a Boolean algebra is associative. 10. Prove that the join-infinite distributive law implies that 2 +3 3−M • 2 ,4 4−N œ 2 Ð+3 • ,4 Ñ Ð3ß4Ñ−M‚N 11. Let F be the collection of all finite unions of intervals of the real line ‘ of the form Ð∞ß +Ñß Ò+ß ,Ñß Ò,ß ∞Ñ together with the empty set. Show that F is a Boolean algebra under set inclusion but has no atoms. 12. Prove that every Boolean lattice is Brouwerian and Heyting. 13. A lattice P is disjunctive if + Ÿ y , implies that there is an B − P such that B•+! a) and B • , œ ! Formulate the dual concept of being conjunctive. Boolean Algebras 143 b) Show that a Boolean algebra is both disjunctive and conjunctive. For the following exercises, let 0 À E Ä F and 1À F Ä G be functions between nonempty sets. Let 0 ‡ À kÐEÑ Ä kÐFÑ be the induced map defined by 0 ÐWÑ œ Ö0 Ð=Ñ ± = − W× for W − kÐEÑ and let 0 " À kÐFÑ Ä kÐEÑ be the induced inverse map defined by 0 " ÐWÑ œ Ö+ − E ± 0 Ð+Ñ − W× for all W − kÐFÑ, and similarly for 1. Prove the following statements. 14. For all X − kÐFÑ, 0 ‰ 0 " ÐX Ñ © X with equality if and only if 0 is surjective. 15. For all W − kÐEÑ, 0 " ‰ 0 ÐWÑ ª W with equality if and only if 0 is injective. 16. a) 0 is injective (surjective) if and only if 0 " is surjective (injective). b) 0 is injective (surjective) if and only if Ð0 " Ñ" is injective (surjective). 17. Ð1 ‰ 0 Ñ" œ 0 " ‰ 1" 18. 0 " is a Boolean homomorphism, that is, for subsets Y ß Y3 of F , 0 " Š.Y3 ‹ œ .0 " ÐY3 Ñ 0 " Š,Y3 ‹ œ ,0 " ÐY3 Ñ 0 " ÐE Ï Y Ñ œ F Ï 0 " ÐY Ñ 19. The induced map 0 ‡ À kÐEÑ Ä kÐFÑ preserves union, that is, for Y3 © E, 0 ‡ Š.Y3 ‹ œ .0 ‡ ÐY3 Ñ 20. If 0 À E Ä F is a bijection then Ð0 " Ñ" œ 0 ‡‡ Frink's Condition and Distributivity Definition A poset T has Frink's condition if it has finite meets and if there is a function 0 À T Ä T and an element / − T for which ?Ÿ@ Í ? • 0 Ð@Ñ œ / 144 Lattices and Ordered Sets 21. Assume that T is a poset satisfying Frink's condition. Prove the following statements about T : a) / is the smallest element of T , so we will denote it by !. b) B • 0 ÐBÑ œ ! for all B. c) 0 # ÐBÑ Ÿ B for all B. d) 0 is order-reversing: BŸC Í 0 ÐCÑ Ÿ 0 ÐBÑ e) 0 Ð!Ñ is the largest element of T , so we write it as ". f) 0 # ÐBÑ œ B for all B. g) 0 is unique, that is, if any function 1À T Ä T has the same property as 0 then 1 œ 0 . h) T has finite joins. Hence T is a lattice. Look at 0 Ð0 Ð+Ñ • 0 Ð,ÑÑ. i) de Morgan's laws hold in T . j) The function 0 is a complement hence T is a complemented lattice. k) Complements are unique, hence T is a uniquely complemented lattice and we can write 0 ÐBÑ œ Bw . l) Conclude that T is a uniquely complemented lattice in which de Morgan's laws hold and so T is a Boolean lattice. Chapter 6 The Representation of Distributive Lattices We now turn to the issue of representing distributive lattices as power set sublattices. The Representation of Distributive Lattices with DCC The F -down map 9FßÆ À P Ä kÐFÑ is the key to several important representation theorems for distributive lattices. The F -down map is a meethomomorphism, that is, FÆ ÐB • CÑ œ FÆ ÐBÑ ∩ FÆ ÐCÑ for all Bß C − P. Furthermore, 9FßÆ is a join-homomorphism if and only if FÆ ÐB ” CÑ © FÆ ÐBÑ ∪ FÆ ÐCÑ for all Bß C − P, since the reverse inclusion is always true. But this is equivalent to the statement that every , − F is join-prime: , ŸB”C Ê ,ŸB or , Ÿ C Thus, 9FßÆ is a lattice homomorphism if and only if every element of F is joinprime. Also, Theorem 3.32 implies that 9FßÆ is injective if and only if F is joindense in P. Theorem 6.1 Let F be a nonempty subset of a lattice P and let 9FßÆ À P Ä kÐFÑ be the F -down map. Then the following hold: 1) 9FßÆ is a lattice homomorphism if and only if every element of F is joinprime in P. 2) 9FßÆ is injective if and only if F is join-dense in P. 3) 9FßÆ is a lattice embedding if and only if F is a join-dense set of join-prime elements. S. Roman (ed.), Lattices and Ordered Sets, doi: 10.1007/978-0-387-78901-9_6 , © Steven Roman 2008 145 146 Lattices and Ordered Sets Now, if P is distributive, then the join-prime elements are the join-irreducible elements ] ÐPÑ. Moreover, if P also has the DCC, then Theorem 3.29 implies that ] ÐPÑ is join-dense in P. Thus, we have the following representation theorem for distributive lattices with the DCC. Theorem 6.2 (Representation theorem for distributive lattices with DCC) Let P be a distributive lattice with the DCC and let 9] ßÆ À P ä bÐ] Ñ be the ] -down map, where ] œ ] ÐPÑ. 1) 9] ßÆ is a lattice embedding and so P is isomorphic to a power set sublattice of kÐ] Ñ. 2) (Birkhoff [6], 1933) If P is also finite, then 9] ßÆ is a lattice isomorphism and so P ¸ bÐ] Ñ. Proof. For part 2), to see that 9] ßÆ is surjective, we have 9] ßÆ Ð!Ñ œ g. Let H œ Ö." ß á ß .8 × − bÐ] Ñ be a nonempty down-set of join-irreducibles in P and let + œ 1.3 . If 4 − ] satisfies 4 Ÿ +, then 4 Ÿ .5 for some 5 and so 4 − H. Thus, H œ ]Æ Ð+Ñ. Note that by duality, a distributive lattice with the ACC is also isomorphic to a power set sublattice. The Representation of Atomic Boolean Algebras If P is a bounded lattice, then the ] -down map is a Ö!ß "×-homomorphism, since 9] ßÆ Ð!Ñ œ g and 9] ßÆ Ð"Ñ œ ] ÐPÑ Hence, if F is a Boolean algebra, then the ] -down map is a Boolean homomorphism. Moreover, Theorem 5.2 implies that ] ÐFÑ œ TÐFÑ and that TÐFÑ is join-dense (that is, F is atomistic) if and only if F is atomic. Also, since T œ TÐFÑ is an antichain, it follows that bÐTÑ œ kÐTÑ This leads to the following representation theorem for Boolean algebras. Theorem 6.3 (Representation theorem for atomic Boolean algebras) Let F be an atomic Boolean algebra and let 9TßÆ À F Ä kÐTÑ be the T-down map, where T œ TÐFÑ. Representation of Distributive Lattices 147 1) 9TßÆ is a Boolean algebra embedding and so F is isomorphic to a power set subalgebra of kÐTÑ. 2) (Lindenbaum and Tarski, see [62]) If F is complete, then 9FßÆ is a Boolean algebra isomorphism and so F ¸ kÐTÑ. Conversely, any power set is complete and atomic. Thus, up to isomorphism, the complete atomic Boolean algebras are precisely the power set algebras. Proof. For part 2), to see that 9TßÆ is surjective, we have 9TßÆ Ð!Ñ œ g. Let E © T be nonempty. We wish to show that E œ TÆ Ð1EÑ. Clearly, E © TÆ Ð1EÑ. For the reverse inclusion, if α − TÆ Ð1EÑ, then infinite joindistributivity implies that α œ α • Š2E‹ œ 2+−E Ðα • +Ñ Since α is an atom, it follows that α œ α • + for some + − E, that is, α Ÿ +. But + is also an atom and so α œ + − E. We leave proof of the converse to the reader. The Representation of Arbitrary Distributive Lattices Garrett Birkhoff also showed that an arbitrary distributive lattice P is isomorphic to a power set sublattice, using the family of prime ideals as a carrier set. For + − P, let c+ œ ÖT − SpecÐPÑ ± + − T × and let cc+ œ Ðc+ Ñ- œ ÖT − SpecÐPÑ ± +  T × The first order of business is to show that cc+ is nonempty for + !. Theorem 6.4 Let P be a distributive lattice. 1) If + Ÿ y , , then there is a prime ideal containing , but not +. 2) If + !, then cc+ is nonempty. 3) If + Á , , then cc+ Á cc, . Proof. For part 1), since + Â Æ , , the family Y of all ideals containing , but not + is nonempty. Since the union of any chain in Y is in Y , Zorn's lemma implies that Y has a maximal ideal Q . But Q is also prime, for if B • C − Q but Bß C  Q , then Q ” Æ B œ P and Q ” Æ C œ P. Thus, + Ÿ 7 ” B and + Ÿ 8 ” C for some 7ß 8 − Q . Setting : œ 7 ” 8 − Q gives + Ÿ : ” B and + Ÿ : ” C and so + Ÿ Ð: ” BÑ • Ð: ” CÑ œ : ” ÐB • CÑ − Q whence + − Q , a contradiction. Thus, Q is prime. The other parts follow from part 1). 148 Lattices and Ordered Sets It is easy to see that c+ ∩ c, œ c+”, ß c+ ∪ c, œ c+•, and Ðc+ Ñ- œ c+w the latter when + has a complement. Thus cc+ ∩ cc, œ ccÐ+•,Ñ ß cc+ ∪ cc, œ ccÐ+”,Ñ and Ðcc+ Ñ- œ ccÐ+w Ñ the latter when + has a complement. In other words, the set f œ Öcc+ ± + − P× is a sublattice of the power set kÐSpecÐPÑÑ and if P is complemented, then f is a subalgebra of kÐSpecÐPÑÑ. Also, the map 3À P Ä bÐSpecÐPÑÑ defined by 3Ð+Ñ œ cc+ which we will call the rho map for P, is a lattice homomorphism that preserves existing complements. Moreover, Theorem 6.4 shows that the rho map is a lattice embedding. We can now summarize. Theorem 6.5 Let P be a lattice and let f œ Öcc+ ± + − P× 1) f is a sublattice of the power set lattice kÐSpecÐPÑÑ, in fact, cc+ ∩ cc, œ ccÐ+•,Ñ and cc+ ∪ cc, œ ccÐ+”,Ñ Also, if P is complemented, then f is a subalgebra of kÐSpecÐPÑÑ, that is, Ðcc+ Ñ- œ ccÐ+w Ñ 2) The rho map 3À P Ä bÐSpecÐPÑÑ defined by 3Ð+Ñ œ cc+ is a lattice embedding if and only if P is distributive. If P is a finite distributive lattice, then the rho map is also surjective. To see this, Theorem 4.27 says that all prime ideals have the form Ð Å 4Ñ- for 4 − ] ÐPÑ. Let H œ ÖÐ Å 4" Ñ- ß á ß Ð Å 48 Ñ- × − bÐSpecÐPÑÑ We show that H œ cc+ , where + œ 143 . Now, if 4 is join-irreducible, then since Representation of Distributive Lattices 149 H is a down-set, the following are equivalent: Ð Å 4Ñ- − cc+ +  РŠ4Ñ4 Ÿ + œ 245 4 Ÿ 45 for some 5 Ð Å 4Ñ- © Ð Å 45 ÑÐ Å 4Ñ- − H Thus, H œ cc+ and so 3 is surjective and hence a lattice isomorphism. Theorem 6.6 Let P be a nontrivial distributive lattice and let 3À P Ä bÐSpecÐPÑÑ be the rho map for P. 1) (Representation theorem for distributive lattices, Birkhoff, 1933 [6]) The rho map 3 is an embedding of P into the power set kÐSpecÐPÑÑ and so P is isomorphic to a power set sublattice (ring of sets). Moreover, if P is finite, then 3 is an isomorphism and P ¸ bÐSpecÐPÑÑ. 2) (Representation theorem for Boolean algebras) If P is a Boolean algebra, then 3 is a Boolean algebra embedding of P into the power set kÐSpecÐPÑÑ and so P is isomorphic to a power set subalgebra (field of sets). Moreover, if P is finite, then 3 is an isomorphism. In the case of a finite distributive lattice P, the rho map 3À P ¸ bÐSpecÐPÑÑ has image bÐSpecÐPÑÑ. However, to describe the image of the rho map for an arbitrary distributive lattice requires some topological notions and so we postpone this matter until the chapter on duality theory, where the requisite topological notions are developed. Summary Let us summarize our results on lattice representations. Theorem 6.7 (Representation theorems) 1) (Arbitrary lattices) Any lattice P is isomorphic to the lattice c ÐPÑ of principal ideals of P, via the down map 9Æ À P ä \ ÐPÑ. 2) (Distributive lattices) Any distributive lattice P is isomorphic to a power set sublattice (ring of subsets), via the rho map 3À P Ä bÐSpecÐPÑÑ. If P has a chain condition, then the ] -down map 9] ßÆ À P Ä b Ð] Ñ is also an embedding and is an isomorphism if P is finite. 3) (Boolean algebras) Any Boolean algebra F is isomorphic to a power set subalgebra (field of subsets), via the rho map 3À P Ä bÐSpecÐFÑÑ. 150 Lattices and Ordered Sets 4) (Complete atomic Boolean algebras) Any complete atomic Boolean algebra F is isomorphic to the power set kÐTÑ, via the T-down map 9TßÆ À F Ä kÐTÑ. Conversely, any power set is complete and atomic. Exercises 1. Let P be a nontrivial lattice and let f œ Öcc+ ± + − P× 2. 3. 4. 5. Prove that f is a sublattice of the power set sublattice kÐSpecÐPÑÑ, in particular, a) ccÐ+•,Ñ œ cc+ ∩ cc, b) ccÐ+”,Ñ œ cc+ ∪ cc, Prove that if P is complemented then f is a subalgebra of kÐSpecÐPÑÑ, in particular, œ ccÐ+w Ñ c) cc+ Let P be a distributive lattice with the DCC. Show that P can be embedded in a Boolean lattice F . If 8 œ kPk is finite, then show that F can be chosen with kF k Ÿ #8" . Find a lattice P for which ] ÐPÑ œ TÐPÑ but P is not a Boolean algebra. Let T be a finite poset. a) Describe the covering relation in the lattice bÐT Ñ. b) What is the length of bÐT Ñ. c) Show that the length of a finite distributive lattice P is equal to k] ÐPÑk. Let P be a lattice with the DCC. Show that 3 œ 9cÅ ‰ 9] ßÆ 6. where 3 is the rho map, 9cÅ is the not-up map and ] œ ] ÐPÑ. Show that the map 3` À P Ä h ÐSpecÐPÑÑ defined by 3` Ð+Ñ œ c+ œ ÖT − SpecÐPÑ ± + − T × is a lattice anti-homomorphism, that is, 3` Ð+ • ,Ñ œ 3` Ð+Ñ ∪ 3` Ð,Ñ 3` Ð+ ” ,Ñ3` Ð+Ñ ∩ 3` Ð,Ñ and if P is Boolean then 3` is a Boolean algebra anti-homomorphism, that is, 3` Ð+w Ñ œ 3` Ð+Ñ- 7. Let P be a lattice and let c+ ³ Ðcc+ Ñ- œ ÖT − SpecÐPÑ ± + − T × and Representation of Distributive Lattices 151 cc+ß, ³ cc+ ∩ c, Let f œ Öcc+ ± + − P× and U œ Öcc+ß, ± +ß , − P× Prove that U contains f and is closed under intersection and complement, in particular, a) cc+ œ cc+ß! and c, œ cc"ß, b) cc+" ß," ∩ cc+# ß,# œ ccÐ+" •+# ÑßÐ," ”,# Ñ c) cc+ß, œ cc,ß+ Prove that if P is complemented then cc+ß, œ ccÐ+•,w Ñ 8. and so U œ f . Let P be a lattice. We define, for any ideal M of P, ccM ³ ÖT − SpecÐPÑ ± M © y T× Prove that if Mß N ü P then M©N Í ccM © ccN Also, for any + − P, ccÐÆ+Ñ œ cc+ and for any nonempty subset \ © P, . ccB œ ccÐ\Ó B−\ Chapter 7 Algebraic Lattices Motivation We have seen in Theorem 3.8 that the meet-structures in a complete lattice P are the families of closed sets of the closure operators on P. Specifically, the intersection-structures on a nonempty set \ are the families of closed sets of the closure operators on \ . We have also seen that there are two types of closure operators on \ : finitary and infinitary. A finitary closure operator is one for which the closure operator is completely determined by its behavior on finite sets, in the sense that clÐWÑ œ .ÖclÐJ Ñ ± J © Wß J finite× The closure operators one meets (no pun intended) in algebra tend to be finitary, but as we remarked earlier, topological closure operators on X" spaces are finitary if and only if the topology is discrete. In a sense that we will make more precise, the property of being finitary is the dividing line between closure operators that come from algebraic structures and other types of closure operators. To explain this more clearly, we define the concept of a “general algebraic structure,” which encompasses some of the common algebraic structures, such as groups, rings and lattices (but not fields). Definition 1) If 8 is a positive integer, an 8-ary operation on a nonempty set E is a map 0 À E8 Ä E. A !-ary operation on E is a distinguished element of E. The !ary, "-ary and #-ary operations are also called nullary, unary and binary operations, respectively. An operation that is 8-ary for some 8 ! is said to be a finitary operation. (One can also define infinitary ,-ary operations 0 À E, Ä E, where , is any infinite cardinal.) S. Roman (ed.), Lattices and Ordered Sets, doi: 10.1007/978-0-387-78901-9_7, © Steven Roman 2008 153 154 Lattices and Ordered Sets 2) An H-algebra (or just an algebra), also called a universal algebra, is a set E, together with a set H of finitary operations on E. We denote an algebra by ØEà HÙ, or simply by E when the set of operators is understood. 3) Let ØEà HÙ be an algebra. If a subset ] © E is invariant under 0 − H, that is, if C" ß á ß C 8 − ] Ê 0 ÐC" ß á ß C8 Ñ − ] then the restriction 0 l ] À ] 8 Ä ] is an 8-ary operation on ] . (A set ] is invariant under a nullary operation B − E if B − ] .) If ] is 0 -invariant for all 0 − H, then the set ] , together with the restrictions Ö0 l] ± 0 − H× is a subalgebra of ØEà HÙ. When the operations are understood, we simply refer to ] as a subalgebra of E. Let SubÐØEà HÙÑ denote the family of subalgebras of an algebra ØEà HÙ. To clarify an earlier remark, a field is not an algebra, since the inverse operation is only a partial operation: ! has no inverse. The family SubÐEÑ of subalgebras of an algebra E is easily seen to be an ∩ structure, called a subalgebra lattice. Note also that the empty set is a subalgebra of E if and only if H has no nullary operations. If E is an algebra and W © E, then the subalgebra generated by W is the smallest subalgebra of E containing W , which is the intersection of all subalgebras containing W . The subalgebra generated by a finite nonempty set W is said to be finitelygenerated. Our ultimate goal is to characterize the ∩ -structures that are subalgebra lattices. As we will see, these ∩ -structures can be characterized in several ways. In terms of the ∩ -structure itself: 1) They are the ∩ -structures that inherit directed unions from kÐ\Ñ. For example, the directed union of subgroups is a subgroup. Note that this is not true for nondirected (even finite) unions. While these ∩ -structures are most commonly referred to as algebraic ∩ -structures, reflecting their usual Ä origin, we will use the more descriptive name ∩ ∪ -structures. 2) They are the ∩ -structures for which the compact subsets (defined later) are join-dense. For example, every subgroup L of a group is the join of all finitely-generated subgroups of L . In terms of the corresponding closure operator: 3) They are the families of closed sets of finitary closure operators. For example, the “subgroup generated by” operator is finitary. Algebraic Lattices 155 4) They are the families of closed sets of join-continuous closure operators, that is, closure operators with the property that for a directed set H, clŠ..−H . ‹ œ 2.−H clÐ.Ñ A word on notation: The symbols Ä 2H and 2 H Ä denote the join of a directed subset H, referred to as a directed join. Note that a finite directed set has a maximum element. Hence, the join of a finite directed set is equal to the maximum element. Algebraic Lattices The finitely-generated subalgebras of an algebra E have a special property, namely, if a finitely-generated subalgebra W is contained in the join of an arbitrary family Y of subalgebras, then W is contained in the join of a finite subfamily of Y , that is, W © 2Y Ê W © 2 Y! for some finite subfamily Y! of Y . This motivates the following definitions. Definition Let P be a complete lattice and let + − P. 1) A join-cover of + is a subset F of P for which + Ÿ 2F 2) The element + is compact if every join-cover of + has a finite join-subcover, that is, + Ÿ 2F Ê + Ÿ 2F! for some finite subset F! © F The set of compact elements of P is denoted by ^ÐPÑÞ Theorem 7.1 Let P be a complete lattice. An element + − P is compact if and only if every directed join-cover of + has a finite (therefore singleton) joinsubcover, that is, Ä + Ÿ 2 H Ê + Ÿ . for some . − H Proof. It is clear that compact elements have this property. Conversely, suppose that this property holds and + Ÿ 1F for some set F , then + œ 2F œ 2š2W ¹ W © F , W finite› 156 Lattices and Ordered Sets But the set Ö W ± W © F , W finite× is directed and so + Ÿ subset W © F . W for some finite Example 7.2 1) All elements of a finite lattice are compact. 2) The compact elements of kÐ\Ñ are the finite subsets of \ . 3) The compact elements of the subgroup lattice fÐKÑ of a group K are the finitely-generated subgroups. 4) The compact elements of the subspace lattice fÐZ Ñ of a vector space Z are the finite-dimensional subspaces. Theorem 7.3 Let P be a complete lattice. 1) ! − P is compact. 2) The compact elements ^ÐPÑ inherit finite joins from P and so ^ÐPÑ is a join-semilattice with !. 3) The ideals of ^ÐPÑ are precisely the sets of the form ^Æ Ð+Ñ for + − P. Proof. For part 3), let M − \ and let + œ 1Ä M . Then M © ^Æ Ð+Ñ. On the other hand, if B − ^Æ Ð+Ñ, then B is compact and B Ÿ 1Ä M and so B Ÿ 3 for some 3 − M , whence B − M . Thus, M œ ^Æ Ð+Ñ. Definition A complete lattice P is algebraic, or compactly-generated if the compact elements of P are join-dense in P. Note that not all authors require completeness in the definition of an algebraic lattice. All finite lattices are algebraic. We have just seen that the subalgebra lattice SubÐEÑ of an algebra is algebraic and we will show that every algebraic lattice is the subalgebra lattice for some algebra. Ä ∩ ∪ -Structures Subalgebra lattices are ∩ -structures but, in general, they do not inherit arbitrary unions from kÐ\Ñ. However, they do inherit directed unions. This follows from the finitary nature of the operations in an algebra. For if Y œ ÖW3 ± 3 − M× is a directed subfamily of f and if Y œ -Ä Y , then B" ß á ß B 8 − Y Ê B" ß á ß B8 − J for some J − Y and so 0 ÐB" ß á ß B8 Ñ − J © Y for all 0 − H. Hence, Y is a subalgebra of \ . The following definition is motivated by this example. Definition If a nonempty subset ` of a power set kÐ\Ñ inherits arbitrary Ä meets and directed joins, we call ` an ∩ ∪ -structure. Algebraic Lattices 157 Ä Some authors refer to ∩ ∪ -structures as algebraic intersection-structures, but we prefer to avoid this terminology, since intersection-structures are complete lattices and the term algebraic has a different definition when applied to complete lattices (as we will see a bit later). Ä It is not hard to characterize the compact subsets of an ∩ ∪ -structure ` in \ Ä and to show that an ∩ ∪ -structure is an algebraic lattice. Recall that the `closure of a subset W of \ is defined by ØWÙ œ ,ÖQ − ` ± W © Q × Ä Definition Let ` be an ∩ ∪ -structure in \ . A set in ` that is the `-closure ØWÙ of a finite subset W of \ is called a finitely-generated element of `. Any finitely-generated element ØJ Ù of ` is compact, for if Ä ØJ Ù © . ÖQ3 ± Q3 − `ß 3 − M× then J is also contained in this directed union and so J © Q4 for some 4 − M . Hence, ØJ Ù © Q4 . Conversely, suppose that O is compact in `. The family of all `-closures ØJ Ù of finite subsets of O is directed, for if J" and J# are finite, then ØJ" Ùß ØJ# Ù © ØJ" ∪ J# Ù. Hence, Ä O © . ÖØJ Ù ± J © Oß J finite× and so there is a finite subset J of O for which O © ØJ Ù. Since the reverse inclusion is clear, we have O œ ØJ Ù. Ä Theorem 7.4 Let ` be an ∩ ∪ -structure in kÐ\Ñ. 1) The compact elements of ` are the finitely-generated elements of `. 2) ` is an algebraic lattice. Proof. The last statement follows from the fact that for any Q − `, Ä Q œ . ÖØJ Ù ± J © Q ß J finite× Ä Thus, ∩ ∪ -structures are algebraic lattices. On the other hand, every algebraic Ä lattice is isomorphic to an ∩ ∪ -structure. To see this, we show that every algebraic lattice is isomorphic to the ideal lattices of a join-semilattice with ! Ä and that these lattices are ∩ ∪ -structures. Note that the notion of an ideal makes sense in a join-semilattice P: An ideal is a nonempty down-set that inherits binary joins from P. Theorem 7.5 An algebraic lattice P is isomorphic to the ideal lattice of a joinsemilattice with !. In particular, if ^ œ ^ÐPÑ, then the ^-down map 158 Lattices and Ordered Sets 9^ßÆ À P ¸ \ Ð^ÐPÑÑ defined by 9^ßÆ Ð+Ñ œ ^Æ Ð+Ñ Ä is an isomorphism. Moreover, \ Ð^ÐPÑÑ is an ∩ ∪ -structure in kÐ^ÐPÑÑ and so Ä every algebraic lattice is isomorphic to an ∩ ∪ -structure. Proof. Theorem 7.3 implies that ^ÐPÑ is a join-semilattice with ! and that \ œ \ Ð^ÐPÑÑ is an ∩ -structure whose elements are the sets ^Æ Ð+Ñ for + − P. Hence, the ^-down map 9^ßÆ À P Ä \ is surjective. If P is algebraic, then ^ÐPÑ is join-dense in P and so Theorem 3.32 implies that 9^ßÆ is an order embedding and therefore a lattice isomorphism. To see that \ has directed unions, let Ö^Æ Ð.Ñ ± . − H× be a directed family in \ . Since 9^ßÆ is an order isomorphism, it follows that H is also directed. Moreover, .Ö^Æ Ð.Ñ ± . − H× œ Ö5 − ^ÐPÑ ± 5 Ÿ . for some . − H× Ä œ Ö5 − ^ÐPÑ ± 5 Ÿ 2 H× Ä œ ^Æ Š2H‹ Ä and so \ has directed unions and is an ∩ ∪ -structure. Algebraic Closure Operators Ä As we mentioned earlier, ∩ ∪ -structures correspond to finitary closure operators. Such operators have another name. Definition A finitary closure operator on a set \ is called an algebraic closure operator. Algebraic closure operators are continuous in the following sense. Definition Let T and U be posets that have directed joins. A function 0 À T Ä U is continuous (also called join-continuous) if for any directed subset H in T , the set 0 ÐHÑ is also directed and Ä 0 Œ 2 .−H . Ä œ 2 .−H 0 Ð.Ñ A continuous map is monotone, for if + Ÿ ,, then the set Ö+ß ,× is directed and so 0 Ð,Ñ œ 0 Ð+ ” ,Ñ œ 0 Ð+Ñ ” 0 Ð,Ñ Algebraic Lattices 159 which implies that 0 Ð+Ñ Ÿ 0 Ð,Ñ. The converse is not true, however, but we can say that any monotone map sends directed sets to directed sets and that Ä 2 Ä 2 0 Ð.Ñ Ÿ 0 . Œ .−H .−H Example 7.6 Let 0 À kÐÑ Ä kÐÑ be defined by 0 ÐWÑ œ g for W a finite set, 0 ÐWÑ œ otherwise. Then 0 is monotone but not continuous, since if k! ÐÑ is the family of all finite subsets of , then 0 Ð. k! ÐÑÑ œ 0 ÐÑ œ but . 0 Ðk! ÐÑÑ œ . g œ g Theorem 7.7 A closure operator on a set \ is algebraic if and only if it is continuous. Proof. Suppose that cl is algebraic and H œ Ö.3 ± 3 − M× is directed in kÐ\Ñ. Then ÖclÐ.3 Ñ ± 3 − M× is also directed, since clÐ.3 Ñß clÐ.4 Ñ Ÿ clÐ.3 ” .4 Ñ. Also, Ä clŒ . .3 Ä Ä œ . œclÐBÑ ¹ B finiteß B © . .3 Ä œ . šclÐBÑ ¹ B © .3 for some .3 − H› Ä © . eclÐ.3 Ñ ± .3 − Hf and since the reverse inclusion is clear, we have equality and so cl is joincontinuous. Conversely, if cl is join-continuous, then for any + − P, Ä + œ . Ö5 ± 5 finiteß 5 © +× and so join-continuity gives Ä clÐ+Ñ œ . ÖclÐ5Ñ ± 5 finiteß 5 © +× which shows that cl is algebraic. The Main Correspondence If P is a complete lattice, then we have seen in an earlier chapter that the maps >À cl Ä ClÐ\Ñ and CÀ G Ä G -closure are inverse bijections between closure operators on a set \ and ∩ -structures on kÐ\Ñ. As pictured in Figure 7.1, under these bijections, the algebraic closure Ä operators correspond to the ∩ ∪ -structures. 160 Lattices and Ordered Sets Algebraic closure operators C-closure Closed sets Closure operators → ∩∪-Structures ∩-Structures Figure 7.1 Theorem 7.8 Let \ be a nonempty set. The maps >À cl Ä ClÐ\Ñ and CÀ G Ä G -closure are inverse bijections between the closure operators on \ and the ∩ -structures Ä on kÐ\Ñ. These bijections map algebraic closure operators onto ∩ ∪ structures. Proof. Suppose that cl is algebraic and that H œ Ö.3 ± 3 − M× is directed in ClÐ\Ñ. Since cl is join-continuous, Ä clŒ . .3 Ä Ä œ . clÐ.3 Ñ œ . .3 Ä Hence, -Ä H − ClÐ\Ñ and so ClÐ\Ñ is an ∩ ∪ -structure. Conversely, if cl is join-continuous and if + − kÐ\Ñ, then Ä Ä + œ . ÖB ± B finiteß B © +× © . ÖclÐBÑ ± B finiteß B © +× © clÐ+Ñ But the element Ä , œ . ÖclÐBÑ ± B finiteß B © +× is closed by the join-continuity of the closure operator and so taking closures gives clÐ+Ñ œ , and so cl is algebraic. Subalgebra Lattices We can now characterize subalgebra lattices. Theorem 7.9 (Birkhoff–Frink, 1948 [8]) A lattice P is isomorphic to the subalgebra lattice SubÐØEß HÙÑ of some algebra ØEß HÙ if and only if P is algebraic. Ä Proof. Since a subalgebra lattice is an ∩ ∪ -structure, it is algebraic. Let us give two proofs that algebraic lattices are isomorphic to subalgebra lattices. First, Theorem 7.5 implies that P is isomorphic to the ideal lattice \ ÐN Ñ of the join- Algebraic Lattices 161 semilattice N œ ^ÐPÑ with !. Thus, all we need to do is find operations H on N so that the subalgebras of the algebra ØN à HÙ are precisely the ideals of N . At the start, N is an algebra with no operations and so every subset is a subalgebra. To reduce this family to the ideals of N , we first include the binary join operation and so SubÐN Ñ is now the family of subsets of N that inherit binary joins. Ideals are also down-sets and to restrict the subalgebras to downsets, that is, to force B − Wß + Ÿ B Ê +−W for every subalgebra W , we include a unary operation 0+ ÐBÑ œ œ + B if + Ÿ B otherwise for each + − N . Then B − W and + Ÿ B implies that + œ 0+ ÐBÑ − W . Now SubÐN Ñ is the family of all down-sets that inherit binary joins. These are precisely the ideals of N with one exception. Our definition of ideal requires that ideals be nonempty. Thus, we include the nullary operation ! − N to remove the empty set from SubÐN Ñ. Now we have SubÐØN ß HÙÑ œ \ ÐN Ñ ¸ P Ä As a second proof, P is isomorphic to an ∩ ∪ -structure ` on a set \ and so ` is the family ClÐ\Ñ of closed sets of a finitary closure operator cl on \ . Thus, for any W © \ , clÐWÑ œ .ÖclÐJ Ñ ± J © Wß J finite× and so W is closed if and only if it has the property that J © Wß J finite Ê clÐJ Ñ © W (7.10) We seek operations on \ for which this characterizes the subalgebras as well. Given a finite set J © \ of size 8 ! and an + − clÐJ Ñ Ï J , we can force a subalgebra W of \ that contains J to also contain + by requiring that W be invariant under any 8-ary operation 0Ð+ßJ Ñ satisfying 0Ð+ßJ Ñ ÐB" ß á ß B8 Ñ œ + if J œ ÖB" ß á ß B8 × when 8 " and under the nullary operation 0Ð+ßgÑ œ + when 8 œ !. But we also need W to be 0Ð+ßJ Ñ -invariant even if W does not contain J . Of course, this is only an issue when J is nonempty, that is, when 8 ". In this case, 0Ð+ßJ Ñ ÐB" ß á ß B8 Ñ can be any member of W , so we set 162 Lattices and Ordered Sets 0Ð+ßJ Ñ ÐB" ß á ß B8 Ñ œ œ + B" if J œ ÖB" ß á ß B8 × otherwise Then a nonempty subset W of \ is H-invariant, that is, W is a subalgebra of \ if and only if (7.10) holds. Congruence Lattices We have seen (Theorem 7.9) that the subalgebra lattices of algebras are precisely the algebraic lattices. There has been much work done on the question of characterizing the congruence lattices of an algebra. Definition A congruence relation on an algebra Ø\à HÙ is an equivalence relation ´ on \ with the property that for each 8-ary operation 0 − H and for all B3 ß C3 − \ , B3 ´ C3 for " Ÿ 3 Ÿ 8 Ê 0 ÐB" ß á ß B8 Ñ ´ 0 ÐC" ß á ß C8 Ñ The family of all congruence relations on Ø\à HÙ is denoted by ConÐØ\à HÙÑ or ConÐ\Ñ. The meet and join of congruence relations is again a congruence relation and so ConÐ\Ñ is a complete sublattice of the lattice of equivalence relations on \ . The lattice ConÐ\Ñ is called the congruence lattice of \ . We will study congruence lattices in some detail in a later chapter. In 1963, Grätzer and Schmidt proved that the congruence lattices of algebras are also precisely the algebraic lattices. Theorem 7.11 (Grätzer and Schmidt, 1963 [27]) A lattice P is isomorphic to the congruence lattice ConÐØEß HÙÑ of some algebra ØEß HÙ if and only if P is algebraic. The proof of Theorem 7.11 is rather difficult and several proofs have appeared as time has gone by (some published and others not). However, as Grätzer [26] points out in his recent survey article, the proof is somehow inherently complex, as witnessed by the work of Freese, Lampe and Taylor. Theorem 7.12 (Freese, Lampe, and Taylor, 1979 [20]) Let Z be an infinitedimensional vector space whose base field has uncountable cardinality , and let SubÐZ Ñ be the subalgebra lattice of Z , which is algebraic. If SubÐZ Ñ is isomorphic to a congruence lattice of an algebra ØEà HÙ, then kHk ,, that is, Ø\ß HÙ has at least , operations. It is also natural to ask in particular whether every algebraic lattice is isomorphic to the congruence lattice ConÐPÑ of some lattice P. (A latttice is an Algebraic Lattices 163 algebra.) In a later chapter, we will study the congruence lattice ConÐPÑ of a lattice P. We will show that ConÐPÑ is distributive and this brings us to one of the most important and longstanding problems in lattice theory: The congruence lattice problem (CLP) is the problem of whether all distributive, algebraic lattices are isomorphic to ConÐPÑ for some lattice P. For the finite case, we have the following positive result. Theorem 7.13 1) (Dilworth) Every finite distributive lattice is isomorphic to ConÐPÑ for some finite lattice P. 2) (Grätzer and Schmidt, 1963 [27]) Every finite distributive lattice is isomorphic to ConÐPÑ for some finite sectionally complemented lattice P. There is a modestly amusing story behind the theorem of Dilworth stated above. Apparently, Dilworth did not publish this theorem, but it did appear as an exercise (marked with an asterisk to indicate that it is hard) in Birkhoff's book [5]. As Grätzer states in [26]: E. T. Schmidt and I got really interested in the result and inquired from G. Birkhoff where the result came from, but he did not know and encouraged us to write to R. P. Dilworth. Unfortunately, Dilworth was busy editing the proceedings of a lattice theory meeting, but eventually we got a response. Yes, he proved the result, and the proof was in his lecture notes. No, copies of his lecture notes were no longer available. On the other hand, the general congruence lattice problem has very recently (2006) been solved in the negative by Wehrung [64]. Meet-Representations We have defined a meet-irreducible element 7 − P to be a nonunit element with the property that 7œ+•, Ê 7 œ + or 7 œ , We can define an analogous notion for arbitrary meets. Definition Let P be a complete lattice. 1) An element + − P is completely meet-irreducible if + œ 3W for a subset W © P implies that + œ = for some = − W . 2) An element + − P is strictly meet-irreducible if + œ " or if the set Å + Ï Ö+× of elements properly containing + has a smallest element. We leave it as an exercise to show that these two concepts are equivalent. 164 Lattices and Ordered Sets Theorem 7.14 An element of a complete lattice is completely meet-irreducible if and only if it is strictly meet-irreducible. Example 7.15 To illustrate the concept, let W be a proper subspace of a vector space Z and let W § œ Ð Å WÑ Ï ÖW×. If @  W , then the subspace ØWß @Ù covers W and so is a minimal element of W § . Hence, W is strictly meet-irreducible if and only if W œ Z or W has codimension ". Note also that any subspace of Z is the intersection of strictly meet-irreducible subspaces. Example 7.16 We can also find strictly meet-irreducible subgroups of a group K as follows. Let L be a proper subgroup of K and let 5  L . Let Y be the family of all subgroups of K that contain L but not 5 . Since the union of any chain in Y is also in Y , Zorn's lemma implies that Y has a maximal member Q . It is easy to see that Q ” Ø5Ù is the smallest element of Q § œ Ð Å Q Ñ Ï ÖQ ×, for if Q § R Ÿ K, then L © R and the maximality of Q implies that 5 − R and so Q ” Ø5Ù © R . Hence, Q is strictly meet-irreducible. We have shown that given any proper subgroup L of K, for each 5  L there is a strictly meetirreducible subgroup Q5 for which L © Q5 and 5  Q5 . Thus, L œ ,ÖQ5 ± 5  L× that is, the strictly meet-irreducible subgroups are meet-dense in the lattice fÐKÑ of subgroups of K. Definition Let P be a lattice. A representation of an element + − P of the form + œ 4Ö;3 ± 3 − M× where each ;3 is strictly meet-irreducible is called a meet-representation of + and is said to be irredundant if + is not the meet of any proper subfamily of Ö;3 ± 3 − M×. The existence of meet-representations does not guarantee the existence of irredundant meet-representations, as the following example shows. Example 7.17 The set P œ Ö!ß "ß "Î#ß "Î$ß á × © is a chain under the usual order. It is clear that every positive element of P is strictly meet-irreducible, but ! is not. Also, ! has a meet-representation but no irredundant meetrepresentation. Theorem 7.18 If P is an algebraic lattice, then every element of P has a meetrepresentation. Proof. The proof mimics the case of groups described earlier, where the role of the element 5 − K is played by a compact element of P. Let +ß , − P satisfy ,Ÿ y +. Since ^ÐPÑ is join-dense, there is a 5 − ^ÐPÑ for which 5 Ÿ , but Algebraic Lattices 165 5Ÿ y +. Let \ œ ÖB − P ± + Ÿ Bß 5 Ÿ y B× which is nonempty since + − \ . Any directed subset H of \ has a join 4 œ 1H, which is also in \ since + Ÿ 4 and if 5 Ÿ 4, then the compactness of 5 implies that 5 Ÿ . for some . − H, which is false. Thus, Zorn's lemma implies that \ has a maximal element 7. It follows that + Ÿ 7 but 5 Ÿ y 7 and so 7 7 ” 5 . Moreover, if 7 8, then the maximality of 7 implies that 5 Ÿ 8 and so 7 ” 5 Ÿ 8. It follows that 7 ” 5 is the least element of Ð Å 7Ñ Ï Ö7× and so 7 is strictly meet-irreducible. Now, if + − P, then for each , − P with the property that , Ÿ y +, there is a strictly meet-irreducible element 7, for which + Ÿ 7, but , Ÿ y 7, . Let ` be the family of all strictly meet-irreducible elements of P that contain + and let , œ 4` Then + Ÿ , but if + , , then there is a strictly meet-irreducible element 7, for which + Ÿ 7, but , Ÿ y 7, . But 7, − ` implies that , Ÿ 7, , a contradiction. Hence, + œ , is the meet of strictly meet-irreducible elements of P. The book by Crawley and Dilworth [11] contains many results relating the existence of irredundant meet-representations to strong atomicity. Here is a sampling. Theorem 7.19 Let P be an algebraic lattice. 1) If P is strongly atomic, then every element of P has an irredundant meetrepresentation. 2) By way of converse, if P is modular and every element of P has an irredundant meet-representation, then P is strongly atomic. It is interesting to note that the statement 1) If P is a strongly atomic algebraic lattice, then every element of P has an irredundant meet-representation is actually equivalent to the axiom of choice. Crawley and Dilworth's proof of this statement uses the Hausdorff maximality principle, which is equivalent to the axiom of choice. On the other hand, suppose that 1) holds and let T be a poset. We may assume that T is not a chain. Let P be the family of all chains in T , along with the Ä empty set and T . Then P is an ∩ ∪ -structure and therefore an algebraic lattice. Also, the strictly meet-irreducible elements in P are T , the maximal chains and the chains that are covered by a maximal chain. If G § H in P, then there is a . − H Ï G and so G ∪ Ö.× is a chain that covers G . Hence, P is strongly atomic 166 Lattices and Ordered Sets and so 1) implies that every proper chain in T is the meet of maximal chains and thus contained in a maximal chain. Thus, the Hausdorff maximality principle holds. Exercises 1. Prove that GÐ:Ñ is the smallest closed element containing : and that " is closed. 2. Prove that the compact elements ^ÐPÑ in a complete lattice P inherit finite joins from P and so ^ÐPÑ is a join-semilattice with !. 3. Is the meet of compact elements necessarily compact? 4. Prove that if P is a complete lattice with ACC, then P is algebraic. 5. Prove that the compact elements of kÐ\Ñ are the finite subsets of \ . 6. Find the compact elements of the chain Ò!ß "Ó © ‘. 7. Find the compact elements of the lattice Ðß ± Ñ. 8. Prove that the compact elements of the subgroup lattice of a group K are the finitely-generated subgroups. 9. Let P be an algebraic lattice. Show that if + 5 and 5 − ^ÐPÑ, then 5 is a cover for some element , for which + Ÿ , 5 . 10. Let P be an algebraic lattice and let W be a complete sublattice of P. Show that W is also algebraic. Conclude that any closed interval of P is algebraic. 11. Prove that any algebraic lattice is weakly atomic. 12. Let P be an algebraic lattice. a) Prove that P is meet-continuous, that is, meet distributes over arbitrary directed joins, that is, Ä Ä + • 2 Ö,3 ± 3 − M× œ 2 Ö+ • ,3 ± 3 − M× b) Prove that if P is distributive, then meet distributes over arbitrary join. 13. Prove that an element + − P is completely meet-irreducible if and only if it is strictly meet-irreducible. 14. Let P be a lattice. An element + − P is join-inaccessible if for any subset F of P, + œ 2F Ê + œ 2F! for some finite subset F! © F Prove the following for a complete lattice P. a) The compact elements of P are join-inaccessible. b) If P is algebraic, then the compact elements are precisely the joininaccessible elements. 15. Let P be a finite lattice in which every element has a unique irredundant meet-representation. Let W œ Ö=3 ± 3 − M× be the family of strictly meetirreducibe elements of P. Thus, for each + − P, there is a unique subset WÐ+Ñ © W for which Algebraic Lattices 167 + œ 43−WÐ+Ñ =3 is an irredundant meet-representation of +. a) Show that + œ , if and only if WÐ+Ñ œ WÐ,Ñ. b) Show that + , if and only if WÐ,Ñ § WÐ+Ñ. c) Show that + ö , if and only if WÐ,Ñ ö WÐ+Ñ. d) Show that - œ 43−WÐ+Ñ∩WÐ,Ñ =3 is an irredundant meet-representation of + ” , . e) Show that P is upper semimodular. 16. Prove the following (unpublished?) result of Bill Hanf: For any nontrivial algebraic lattice P, the following are equivalent: a) If 5 − ^ÐPÑ is compact, then ^Æ Ð5Ñ is countable. b) P is isomorphic to the subalgebra lattice SubÐTÑ of an algebra T œ Ø\ß HÙ that has only a countable number of operations (including nullary operations). c) P is isomorphic to the subalgebra lattice SubÐTÑ of an algebra T œ Ø\ß HÙ that has only a single operation, which is binary. Chapter 8 Prime and Maximal Ideals; Separation Theorems Separation Theorems The existence of maximal and prime ideals is an important topic in lattice theory. In discussing the existence of such ideals, it is of considerable interest to know not just when a lattice P has such an ideal, but when P has such an ideal that “separates” certain elements or subsets of P. Theorems to this effect are called separation theorems. The following notation will prove convenient. Definition Let P be a lattice. If Eß \ © P are disjoint subsets of P, then we denote the set of all proper ideals of P that contain E and are disjoint from \ by \ ÐPà Eß \Ñ, that is, \ ÐPà Eß \Ñ œ ÖN − \ ÐPÑ Ï ÖP× ± E © N ß N ∩ \ œ g× If M is a proper ideal and J is a proper filter, we denote this set by \ ÐPà M – Pß J — PÑ An element N − \ ÐPà Eß \Ñ represents a type of separation of E and \ and it will be convenient to establish some general terminology to describe this separation. Definition Let Y and Z represent families of subsets of lattices, such as the family of proper subsets, proper ideals or proper filters, ordered by set inclusion. 1) A lattice P has the maximal ÐY ß Z Ñ-separation property if for any disjoint J − Y and K − Z , the set \ ÐPà J ß KÑ has a maximal element. 2) A lattice P has the prime ÐY ß Z Ñ-separation property if for any disjoint J − Y and K − Z , the set \ ÐPà J ß KÑ contains a prime ideal of P. There are some special cases of these definitions that are of interest. S. Roman (ed.), Lattices and Ordered Sets, doi: 10.1007/978-0-387-78901-9_8, © Steven Roman 2008 169 170 Lattices and Ordered Sets Definition 1) (Z œ Ög×) A lattice P has the maximal (prime) Y -extension property if any J − Y is contained in a maximal (prime) ideal of P. 2) (Y œ Ög×) A lattice P has the maximal (prime) Z -avoidance property if for any K − Z , the set \ ÐPà gß KÑ contains a maximal element (prime ideal). 3) (Y œ Z œ Ög×) A lattice P has the maximal (prime) existence property if P contains a maximal (prime) ideal. It is clear that the separation property implies the other properties and all properties imply the existence property. The families of interest to us with regard to separation are the families \ ‡ , Y ‡ and k ‡ of proper ideals, proper filters and proper subsets, respectively and the separations of interest are Ð\ ‡ ß k ‡ Ñ-separation and Ð\ ‡ ß Y ‡ Ñ-separation and those cases formed by replacing one or more of \ ‡ , Y ‡ or k ‡ by the family Ög×. In particular, for a given class V of lattices, we consider the following six separation statements: Ú Ý Ý Ý Ý Ý Ý Every nontrivial V-lattice has the maximal Û Ý Ý Ý Ý Ý Ý Ü Ð\ ‡ ß k ‡ Ñ-separation Ð\ ‡ ß Y ‡ Ñ-separation \ ‡ -extension k ‡ -avoidance Y ‡ -avoidance existence Þ á á á á á á ß property (8.1) á á á á á á à Maximal Separation Theorems for Lattices with " We will prove that if V is a class of lattices with " that contains the class of complete distributive lattices, then each of the six statements in (8.1) is equivalent to the axiom of choice. It is easy to see that the axiom of choice, or equivalently Zorn's lemma, implies the strongest of these statements, and therefore all of these statements: 1) Every nontrivial lattice with " has the maximal Ð\ ‡ ß k ‡ Ñ-separation property. For if M − \ ‡ and \ − k ‡ are disjoint, then \ œ \ ÐPà Mß \Ñ is nonempty, since it contains M . Since the union of any chain V in \ is again in \ ß Zorn's lemma implies that \ has a maximal element. Hence, 1) holds. Prime and Maximal Ideals 171 Actually, there has been much work done proving that various forms of the six statements in (8.1) are equivalent to the axiom of choice. Here are some examples: • • • • • • Every nontrivial lattice with " has the maximal \ ‡ -extension property (Mrówka, 1956 [46]). Every nontrivial distributive lattice with " has the maximal \ ‡ -extension property (Klimowsky, 1958 [38]). Every nontrivial lattice with " has a maximal ideal (Scott, 1954 [55]). Every nontrivial distributive lattice with " has a maximal ideal (Klimowsky, 1958 [38]; see also Rubin and Rubin, 1985 [54], page 101). Every nontrivial complete lattice has a maximal ideal (Banaschewski, 1961 [2]). For each nonempty set \ , every complete sublattice of kÐ\Ñ has a maximal ideal (Bell and Fremlin, 1972 [3]). Of course, the weakest statement of the form (8.1) is 2) Every nontrivial complete distributive lattice has a maximal ideal. In 2003, Herrlich [33] proved that the axiom of choice is equivalent to the following statement: 3) The lattice of closed sets of any topological space has a maximal filter. But it is easy to see that the dual of 2), 2` ) Every nontrivial complete distributive lattice has a maximal filter which is equivalent to 2), implies 3). For the family > of closed subsets of a topological space is a ∩ -structure containing g and therefore a complete lattice. Also, since meet is intersection and finite join is union, it follows that > is distributive. Thus, > is a nontrivial complete distributive lattice and so 2` ) implies 3). Thus, 2) implies the axiom of choice and each of the six statements in (8.1) implies the axiom of choice. The following theorem uses a few basic notions from topology, some of which are covered briefly in an appendix to this book. Theorem 8.2 (Herrlich, 2003 [33]) 1) The following statement is equivalent to the axiom of choice: The lattice of closed sets of any topological space has a maximal filter. 2) Each of the six statements in (8.1) is equivalent to the axiom of choice. Proof. We have seen that the axiom of choice implies statement 2) and therefore also this statement. For the converse, let Ö\3 ± 3 − M× be a nonempty family of 172 Lattices and Ordered Sets nonempty sets. We wish to find a choice function 0 − #\3 . We may use the axioms of set theory without the axiom of choice. This allows us to take unions and cartesian products and to use the axiom schema of comprehension, which says that if W is a set and if T is a property, then we may form the subset E œ ÖB − W ± T ÐBÑ holds× Now, let ∞ be a symbol not in -\3 and let ]3 œ \3 ∪ Ö∞× for each 3 − M . Define a topology on ]3 by taking the closed sets to be the finite subsets of \3 , along with the whole space ]3 . The cartesian product ] œ $3−M ]3 is nonempty, since it contains the constant function 0 À M Ä -]3 defined by 0 Ð3Ñ œ ∞ for all 3 − M , or in more formal notation, 0 œ M ‚ Ö∞× © M ‚ .]3 For each 3 − M , the projection map 13 À ] Ä ]3 is defined by 13 Ð0 Ñ œ 0 Ð3Ñ for all 0 − ] . Also, for each F © ]3 , the cylinder with base F is the set 13" ÐFÑ œ Ö0 − ] ± 0 Ð3Ñ − F× We give ] the product topology 7 , that is, the set of all cylinders with open bases is a subbasis for a topology on ] . Since the projection maps are open, a cylinder is an open subset in the product topology if and only if its base is open. Such cylinders are called open cylinders. The family > of closed subsets of ] is a nontrivial complete distributive lattice and so > has a maximal filter Y , which is also a prime filter by Theorem 4.25. We refer to a cylinder in Y as an Y -cylinder and the base of an Y -cylinder as an Y -base. We wish to show that for each coordinate 3 − M , there is an Y -base that is a finite subset of \3 . For this, it is sufficient to show that there is an Y -base that is a proper subset of ]3 , since any Y -base is closed and so is either ]3 or a finite subset of \3 . However, if there is any J − Y for which 13 ÐJ Ñ is a proper subset of ]3 , then J is contained in the cylinder 13" Ð13 ÐJ ÑÑ and since Y is a filter and 13" Ð13 ÐJ ÑÑ ª J − Y it follows that 13" Ð13 ÐJ ÑÑ is an Y -cylinder whose base is a proper subset of ]3 . This leaves only the possibility that 13 ÐJ Ñ œ ]3 for all J − Y . But this contradicts the maximality of Y . To see this, let B − \3 and let Prime and Maximal Ideals 173 Z œ ÒY ß 13" ÐBÑÑ be the filter generated by Y and the closed cylinder 13" ÐBÑ. The filter Z consists of all closed sets O ª J ∩ 13" ÐBÑ, for some J − Y . But Z is strictly larger than Y , since 13" ÐBÑ is not in Y . Also, Z is a proper filter, since g − Z implies that g œ J ∩ 13" ÐBÑ for some J − Y , which is false since 13 ÐJ Ñ œ ]3 and so there is an 0 − J for which 13 Ð0 Ñ œ B. Thus, we have shown that for each coordinate 3 − M , there is an Y -base that is a finite subset of \3 . Next, we show that there is a unique singleton Y -base ÖB3 × in each \3 . If 13" ÐFÑ is an Y -cylinder with base F œ Ö," ß á ß ,8 × in \3 , then 13" Ð," Ñ ∪ â ∪ 13" Ð,8 Ñ œ 13" ÐFÑ − Y and since Y is prime, at least one 13" Ð,3 Ñ is in Y and so Ö,3 × is also an Y -base in \3 . On the other hand, if Ö-3 × is also an Y -base in \3 with -3 Á ,3 , then g œ 13" Ð,3 Ñ ∩ 13" Ð-3 Ñ − Y which contradicts the maximality of Y . Hence, there is a unique singleton Y base Ö,3 × in each \3 and so the function 0 À M Ä -\3 defined by 0 Ð3Ñ œ ,3 is a choice function for the family Ö\3 ± 3 − M×. The fact that lattices have the maximal Ð\ ‡ ß k ‡ Ñ-separation property is sometimes referred to as the maximal separation theorem for lattices. Similarly, lattices satisfy the maximal extension theorem for proper ideals and the maximal ideal theorem. Maximal Separation Theorems for Boolean Algebras Of course, the class of Boolean algebras does not contain the class of complete distributive lattices and so the previous results do not apply to Boolean algebras. With respect to Boolean algebras, we confine ourselves (for now) to the following two statements: 1) Every nontrivial Boolean algebra has the maximal Ð\ ‡ ß k ‡ Ñ-separation property. 2) Every nontrivial Boolean algebra has the maximal k ‡ -avoidance property. We will prove that each of these statements is equivalent to the axiom of choice. On the other hand, the statement that every nontrivial Boolean algebra has the 174 Lattices and Ordered Sets maximal \ ‡ -extension property is strictly weaker than the axiom of choice. (We will revisit this issue a bit later.) Since a Boolean algebra is a lattice with ", we have seen that the axiom of choice implies statement 1), which implies statement 2). Theorem 8.3 Each of statements 1) and 2) is equivalent to the axiom of choice. Proof. It is sufficient to show that statement 2) implies Tukey's lemma. We begin with a few remarks about families of finite character. First, if Y has finite character and if E − Y , then any subset W of E is also in Y , since all finite subsets of W are finite subsets of E and so belong to Y . Hence, E − Y implies ÆE © Y. Second, let \ be an ideal of kÐ\Ñ contained in Y . Then -\ − Y , for if J © -\ is finite, then J is contained in a union -\! of a finite subset of \ , which is in \ , whence J − \ . Thus, -\ − Y . Moreover, if \ is maximal among all ideals of kÐ\Ñ contained in Y , then -\ is a maximal element of Y . For if -\ § R − Y , then \ © Æ Š.\ ‹ § Æ R © Y Hence, the maximality of \ implies that Æ R œ Y . Thus, Y is an ideal of kÐ\Ñ and so \ œ Y , which is false. Now, to see that 2) implies Tukey's lemma, let Y © kÐ\Ñ be a nonempty family of finite character. Then kÐ\Ñ Ï Y is a proper subset of kÐ\Ñ and so statement 2) implies that there is an ideal \ of kÐ\Ñ that is maximal with respect to avoiding kÐ\Ñ Ï Y , that is, maximal with respect to being contained in Y . Hence, by the previous remarks, -\ is maximal in Y and Tukey's lemma holds. Prime Separation Theorems There are a variety of prime separation theorems that follow from the aciom of choice. Here are some examples: 1) (The distributive prime ideal theorem) Every nontrivial distributive lattice has a prime ideal. 2) (The Boolean prime ideal theorem) Every nontrivial Boolean algebra has a prime (i.e., maximal) ideal. 3) (The Boolean prime \ ‡ -extension theorem) Any proper ideal of a Boolean algebra F can be extended to a prime ideal. It is known that these statements are strictly weaker than the axiom of choice. However, we will not pursue this line of inquiry further. The interested reader may want to consult Rubin and Rubin [53]. Prime and Maximal Ideals 175 Our interest centers upon the fact that, assuming the axiom of choice, separation by prime ideals is equivalent to distributivity. First, we show that for a distributive lattice, a maximal element of \ ÐPà M – Pß J — PÑ is a prime ideal in P. Theorem 8.4 If P is a distributive lattice, then any maximal element of \ ÐPà M – Pß J — PÑ is prime in P. Proof. Let Q be maximal in \ ÐPà Mß J Ñ. Suppose that + • , − Q but that +ß ,  Q . Hence, the ideal N+ œ Q ” Æ + properly contains Q and so intersects J . Thus, + ” 7 − J for some 7 − Q and similarly, , ” 8 − J for some 8 − Q . Hence, if : œ 8 ” 7 − Q , then + ” : and , ” : are in J , whence Ð+ • ,Ñ ” : œ Ð+ ” :Ñ • Ð, ” :Ñ − J But + • , and : are in Q and so Q ∩ J Á g, a contradiction. Thus, Q is prime. We can now show that separation by prime ideals is equivalent to distributivity. Theorem 8.5 The following are equivalent for a nontrivial lattice P (assuming the axiom of choice): 1) P is distributive. 2) (The prime Ð\ ‡ ß Y ‡ Ñ-separation property) The set \ ÐPà M – Pß J — PÑ contains a prime ideal of P. 3) (The prime point-separation property) If , Ÿ y + in P, then there is a prime ideal containing + but not , . In particular, if + Á , , then there is a prime ideal containing one of + or , but not the other. The fact that 1) implies 2) is sometimes called the prime separation theorem. Proof. Assume that P is distributive. Zorn's lemma implies that the set \ ÐPà M – Pß J — PÑ has a maximal ideal Q , which is prime by Theorem 8.4. It is clear that 2) implies 3). Finally, if P is not distributive, then it has a sublattice isomorphic to Q$ or R& , as shown in Figure 8.1. b x b y x z y z a M3 a N5 Figure 8.1 176 Lattices and Ordered Sets Now, suppose that T is a prime ideal containing D . Then B • C œ + − T and so one of B or C is in T . But if C − T , then , œ D ” C − T and so B − T . Thus, D − T implies B − T . But B Ÿ y D and so 3) fails. Exercises 1. 2. 3. 4. 5. 6. 7. 8. 9. Let P be a distributive lattice. Prove that c+ œ cc, if and only if + and , are complements. Let P be a nontrivial bounded lattice. Show that P is distributive if and only if every proper ideal of P is the intersection of prime ideals. Let Q be a distributive lattice and let R be a sublattice of Q . Show that if U − SpecÐR Ñ then there is a Uw − SpecÐQ Ñ such that R ∩ Uw œ U. Let P be a Boolean lattice. Prove that a proper filter J is an ultrafilter if and only if for each + − P, we have + − J or +w − J . Describe the principal ultrafilters of the power set lattice kÐWÑ. Let Y be a proper filter in the power set lattice kÐWÑ. Prove that the following are equivalent: a) J is an ultrafilter. b) If c œ ÖE" ß á ß E8 × is a partition of W then exactly one of the blocks E3 is in Y . c) For every E © W , either E − Y or E- − Y . d) If a subset E of W intersects every subset in Y nontrivially then E − Y . True or false: Distributive lattices with " have the property that if M – P and J — P then \ ÐPà Mß J Ñ has an element that is maximal in P. Let P be a distributive lattice and let Q be a proper sublattice of P. Prove that there are prime ideals T and U of P for which T ∩ Q © U but T © y U. For a Boolean lattice P, prove that P has the maximal separation property for proper ideals and proper filters, that is, the set \ ÐPà M – Pß J — PÑ has a maximal element if and only if this set has an element that is maximal in P. 10. Show that if every nontrivial lattice with " has a maximal ideal then Tukey's lemma holds as follows. Let Y © kÐ\Ñ be a nonempty family of finite character and let Z œ Y ∪ Ö\×. Let E • F œ E ∩ F and let E”F œœ E∪F \ if E ∪ F − Y otherwise Show that Z is a nontrivial lattice with ". 11. In an exercise from an earlier chapter, the reader was asked to prove the following: Let P be a distributive lattice and let M ü P and J ý P. Define a binary relation ) on P by B) C if ÐB ” ?Ñ • @ œ ÐC ” ?Ñ • @ for some ? − M and @ − J Prime and Maximal Ideals 177 a) ) is a congruence relation on P. b) If P is bounded then M © Ò!Ó and J © Ò"Ó. c) ) is proper () § P ‚ P) if and only if M ∩ J œ g. Use this to prove that the prime existence property implies that prime Ð\ ‡ ß Y ‡ Ñ-separation property. Hint: Apply the prime ideal theorem to the quotient PÎ) , which is also a distributive lattice with ". Chapter 9 Congruence Relations on Lattices If W is an algebraic structure, such as a lattice, group, ring or module, then an equivalence relation ´ on W that also preserves the algebraic operations of W is called a congruence relation on W . For example, a congruence relation ´ on a group K is an equivalence relation for which 1) + ´ , Ê +" ´ , " 2) + ´ ,ß +w ´ , w Ê ++w ´ ,, w Now, in elementary algebra, one teaches that there is a correspondence between certain special types of substructures and quotient structures. In the case of groups, for example, there is a correspondence between normal subgroups and quotient groups. A more complete story for groups must include the fact that a subgroup L of a group K is normal if and only if equivalence modulo L : +´, Í +L œ ,L is a congruence relation on K. For if L is normal in K and +L œ +w L and ,L œ , w L , then +,L œ +L, w œ +w L, w œ +w , w L Also, if +L œ ,L , then +" L œ L+" œ L, " œ , " L Conversely, if equivalence modulo L is a congruence relation, then +L œ +w Lß ,L œ , w L Ê +,L œ +w , w L or equivalently, +w − +Lß , w − ,L Ê +w , w − +,L which is equivalent to the equation +L,L œ +,L S. Roman (ed.), Lattices and Ordered Sets, doi: 10.1007/978-0-387-78901-9_9, © Steven Roman 2008 179 180 Lattices and Ordered Sets Setting , œ +" gives +L+" L œ L , which implies that +L+" © L for all + − K and so L is normal in K. Similar statements can be made for rings or modules. However, in most treatments of elementary algebra, the role of the congruence relation is underplayed in favor of the role of the special substructure (normal subgroup, ideal, submodule). On the other hand, in algebras in general, there is no special class of substructures that plays the roles of normal subgroups for groups and ideals for rings and we are left only with the relationship between quotient structures and congruence relations. In particular, this is the case for lattices. Only in distributive, sectionally complemented lattices is there a bijection between quotient lattices and ideals. Congruence Relations on Lattices We begin with the definition of a congruence relation on a lattice. Definition An equivalence relation ) on a lattice P is a congruence relation on P if for all +ß ,ß Bß C − P, +) B and , )C Ê Ð+ • ,Ñ)ÐB • CÑ and Ð+ ” ,Ñ)ÐB ” CÑ The equivalence classes under a congruence relation ) are called congruence classes. The congruence class containing + − P is denoted by Ò+Ó or Ò+Ó) . The set of all congruence classes for ) is denoted by PÎ). The set of all congruence relations on P is denoted by ConÐPÑ. We will use the notations +),ß + ´ ), ) and + ´ , interchangeably throughout the chapter and write + ´ , when the specific congruence is understood. Also, the notation ) 5 +´,´) 5 is shorthand for + ´ , and , ´ - . Note that if ? Ÿ @ and ? ´ @, then all elements in the interval Ò?ß @Ó are congruent, for if B − Ò?ß @Ó, then Bœ@•B´?•Bœ? and so every element of Ò?ß @Ó is congruent to ?. Thus, congruence classes are convex subsets of P. Moreover, if + ´ , , then Congruence Relations on Lattices +•, ´,•, œ, 181 and + ” , ´ , ” , œ , and so all elements of the interval Ò+ • ,ß + ” ,Ó are congruent. Theorem 9.1 Let ) and 5 be congruence relations on a lattice P. 1) The congruence classes of ) are convex sets. 2) For all +ß , − P, +) , Í Ð+ • ,Ñ)Ð+ ” ,Ñ in which case every element of the interval Ò+ • ,ß + ” ,Ó is congruent to +. 3) ) œ 5 if and only if +) , Í +5 , for all + , in P. 4) If . is a binary relation on P satisfying + ., Í Ð+ • ,Ñ.Ð+ ” ,Ñ for all +ß , − P, then ) œ . if and only if +) , Í + ., for all + Ÿ , in P. Proof. For part 4), +) , Í Í Í Ð+ • ,Ñ)Ð+ ” ,Ñ Ð+ • ,Ñ.Ð+ ” ,Ñ + ., Characterizing Congruence Relations The following simple result is very useful. Theorem 9.2 An equivalence relation ´ on a lattice P is a congruence relation if and only if for all +ß ,ß B − P, +´, Ê +•B´,•B and + ” B ´ , ” B Proof. If the stated property holds, then + ´ ,ß B ´ C Ê + • B ´ , • Bß , • B ´ , • C Ê+•B´,•C and similarly for joins. Example 9.3 Let P be a distributive lattice and let > − P. Then the binary relations defined by +$ , if +”>œ,”> 182 Lattices and Ordered Sets and if + ., +•>œ,•> are both congruence relations on P. It is easy to see that these relations are equivalence relations. For $ , it is clear that +$ , Ê Ð+ ” BÑ$ Ð, ” BÑ and for meet, the distributivity of P gives +$ , Ê Ê Ê Ê +”>œ,”> Ð+ ” >Ñ • ÐB ” >Ñ œ Ð, ” >Ñ • ÐB ” >Ñ Ð+ • BÑ ” > œ Ð, • BÑ ” > Ð+ • BÑ$ Ð, • BÑ A similar argument can be made for .. There is another characterization of congruence relations, due to Grätzer and Schmidt, that assumes only a reflexive binary relation. Theorem 9.4 A reflexive binary relation ´ on a lattice P is a congruence relation on P if and only if it satisfies the following properties: 1) For all +ß , − P, +´, Í +•, ´+”, 2) (Comparable transitivity) If + Ÿ , Ÿ - in P, then + ´ ,ß ,´- Ê +´- 3) If + Ÿ , and + ´ , , then +•B´,•B and + ” B ´ , ” B for all B − P. Proof. First note that ´ is symmetric by 1). Next, we show that if + Ÿ , and + ´ , , then all elements in Ò+ß ,Ó are congruent. It is easy to see that any B − Ò+ß ,Ó is congruent to both + and , , for we have +œ+•B´,•BœB and Bœ+”B´,”Bœ, Hence, if Bß C − Ò+ß ,Ó, then B ” C ´ +. But B • C − Ò+ß B ” CÓ and so B • C ´ B ” C, which implies that B ´ C. To establish transitivity, let + ´ , ´ - . Then + • , ´ + ” , and so +•, ´+”, Ê , ´+”, Ê ,”- ´+”,”- Congruence Relations on Lattices 183 Similarly, +•, ´+”, Ê +•, ´, Ê +•,”- ´,•- Hence, +•,•- ´,•- ´,”- ´+”,”and so +•,•- ´+”,”Therefore, all elements in the interval Ò+ • , • -ß + ” , ” -Ó, including + and - , are congruent. Thus, ´ is an equivalence relation. We complete the proof using Theorem 9.2. Suppose that + ´ , . If + and , are comparable, then 3) and the symmetry of the relation imply that +•B´,•B and + ” B ´ , ” B for all B − P. If + ² , , then + • , ´ + ” , and 3) gives Ð+ • ,Ñ • B ´ Ð+ ” ,Ñ • B and Ð+ • ,Ñ ” B ´ Ð+ ” ,Ñ ” B for all B − P. But + • Bß , • B − ÒÐ+ • ,Ñ • Bß Ð+ ” ,Ñ • BÓ and so + • B ´ , • B. Similarly, + ” Bß , ” B − ÒÐ+ • ,Ñ ” Bß Ð+ ” ,Ñ ” BÓ and so + ” B ´ , ” B. Congruence Relations and Partitions It is useful to look at the congruence relations on a lattice P in three ways (we use the same notation for all three viewpoints, relying on the context to make the necessary distinctions): 1) As special types of binary relations on P. 2) As special types of partitions of P. 3) As special subsets of the lattice product P# œ P ‚ P (with coordinatewise meet and join). The first viewpoint provides the definition. For the others, we have the following. Theorem 9.5 1) A partition ) on a lattice P defines a congruence relation on P if and only if the following hold: a) Each block of the partition is a convex sublattice of P. 184 Lattices and Ordered Sets b) The quadrilateral property holds: If two adjacent elements of a quadrilateral U œ Ð+ • ,ß +ß ,ß + ” ,Ñ belong to the same block, then so do the complementary adjacent elements. This is shown in Figure 9.1. a∨b θ θ θ θ a b a∧b Figure 9.1 2) A subset ) of P# is a congruence relation on P if and only if the following hold: a) ) is an equivalence relation on P. b) ) is a sublattice of the lattice P# (under coordinatewise meet and join). Proof. For part 1), one direction has been proved. Assume that 1a) and 1b) hold. We prove that + ´ , implies that + • B ´ , • B for all B − P. Since 1a) and 1b) are self-dual, it follows that + ” B œ , ” B for all B − P. First suppose that ? ´ @ and ? Ÿ @. Then we can project B into the interval Ò?ß @Ó to get ? ” Ð@ • BÑ, as shown in Figure 9.2. v u∨ (v∧ x) x v∧x u u∧ x Figure 9.2 The convexity of blocks implies that ? ´ ? ” Ð@ • BÑ and the quadrilateral property then implies that ? • B ´ @ • B. For the general case, if + ´ , , then letting ? œ + • , and @ œ + ” , , we have + • , ´ + ” , and + • , Ÿ + ” , . Hence, the previous remarks show that Ð+ • ,Ñ • B ´ Ð+ ” ,Ñ • B and since + • B and , • B lie in the interval between Ð+ • ,Ñ • B and Congruence Relations on Lattices 185 Ð+ ” ,Ñ • B, convexity implies that + • B ´ , • B, as desired. We leave proof of part 2) as an exercise. The Lattice of Congruence Relations We have seen in an earlier chapter that if \ is a nonempty set, then the poset EquÐ\Ñ is a complete lattice. When \ is a lattice, we can say a bit more. Let us recall the definition of a witness sequence and add an additional definition. Definition Let Y be a family of binary relations on a lattice P. If +ß , − P, then a witness sequence from + to , is a sequence + œ +" ß +# ß á ß +8" ß +8 œ , in P for which )" )8" )# + œ +" ´ +# ´ +$ ´ â ´ +8" ´ +8 œ , where )3 − Y . When + Ÿ , , an increasing witness sequence from + to , is a witness sequence for which +3 Ÿ +3" for all 3. Ä Theorem 9.6 Let P be a lattice. Then EquÐPÑ is an ∩ ∪ -structure and therefore an algebraic lattice. 1) If ) œ 2Y then +), if and only if there is an increasing witness sequence )3 " )3 # )37" + • , œ B" ´ B# ´ B$ ´ â ´ B7" ´ B7 œ + ” , from + • , to + ” , using Y . 2) ConÐPÑ is a complete sublattice of EquÐPÑ, called the congruence lattice of P. The smallest element of ConÐPÑ is equality and the largest is given by +), for all +ß , − P. Ä 3) ConÐPÑ is an ∩ ∪ -structure and therefore an algebraic lattice. 4) ConÐPÑ is distributive. Ä Proof. We leave the proof that EquÐPÑ is an ∩ ∪ -structure to the reader. For part 1), let ) œ 1Y . If there is an increasing witness sequence from + • , to + ” , , then Ð+ • ,Ñ)Ð+ ” ,Ñ and so +), . Conversely, if +), then Ð+ • ,Ñ)Ð+ ” ,Ñ. Suppose that + • , œ +" ´ +# ´ +$ ´ â ´ +8" ´ +8 œ + ” , is a witness sequence using Y . Write α œ + • , and " œ + ” , . Joining the endpoints of a step +5 ´ +5" with α and then meeting with " gives Ð+5 ” αÑ • " ´ Ð+5" ” αÑ • " 186 Lattices and Ordered Sets where each term is in the interval Ò+ • ,ß + ” ,Ó. Moreover, since this process has no effect on + • , and + ” , , we may assume to start with that +5 − Ò+ • ,ß + ” ,Ó. Then replacing +5 with +" ” â ” +5 gives + • , œ +" ´ + " ” + # ´ + " ” + # ” + $ ´ â â ´ Ð+" ” â ” +8" Ñ ´ Ð+" ” â ” +8 Ñ œ + ” , which is an increasing witness sequence from + • , to + ” , . For part 2), if Y œ Ö)3 − ConÐPÑ ± 3 − M× and if 5 œ +Y , then for any B − P, +5, Ê +)3 , for all 3 Ê Ð+ • BÑ)3 Ð, • BÑ and Ð+ ” BÑ)3 Ð, ” BÑ for all 3 Ê Ð+ • BÑ5Ð, • BÑ and Ð+ ” BÑ5Ð, ” BÑ and so 5 − ConÐPÑ. For the join ) œ 1Y , if + œ +" ß +# ß á ß +8" ß +8 œ , is a witness sequence from + to , , then + • B œ +" • Bß +# • Bß á ß +8" • Bß +8 • B œ , • B is a witness sequence from + • B to , • B and similarly for the join in place of meet. Hence, ) − ConÐPÑ. We leave proof of part 3) as an exercise. For part 4), since ) • Ð7 ” 5 Ñ ª Ð ) ∩ 7 Ñ ” Ð ) ∩ 5 Ñ it is sufficient to prove that +Ò) • Ð7 ” 5ÑÓ, Ê +ÒÐ) ∩ 7 Ñ ” Ð) ∩ 5ÑÓ, for all +ß , − P. The antecedent implies that +), and that there is an increasing witness sequence 7 ∪5 7 ∪5 7 ∪5 + • , œ +" ´ +# ´ +$ ´ â ´ +8" ´ +8 œ + ” , Since Ð+ • ,Ñ)Ð+ ” ,Ñ and since +3 − Ò+ • ,ß + ” ,Ó, it follows that ) ) ) α α + • , œ +" ´ +# ´ +$ ´ â ´ +8" ´ +8 œ + ” , and so if α œ ) ∩ Ð7 ∪ 5Ñ, then α + • , œ +" ´ +# ´ +$ ´ â ´ +8" ´ +8 œ + ” , But α œ Ð) ∩ 7 Ñ ∪ Ð) ∩ 5Ñ and so Ð+ • ,ÑÒÐ) ∩ 7 Ñ ” Ð) ∩ 5ÑÓÐ+ ” ,Ñ which implies that +ÒÐ) ∩ 7 Ñ ” Ð) ∩ 5ÑÓ, . Hence, ConÐPÑ is distributive. Congruence Relations on Lattices 187 Commuting Congruences and Joins We have seen that +Ð) ” 5Ñ, if and only if there is a witness sequence ." .# .8" + œ +" ´ +# ´ +$ ´ â ´ +8" ´ +8 œ , where .3 œ ) or .3 œ 5. By transitivity, we can assume that adjacent .3 's are distinct. Let us focus on one portion of this sequence, say ) 5 B´C´D If it were true that ) and 5 commute, even at the cost of changing the intermediate element C, that is, if ) 5 B´C´D 5 Ê ) B´A´D for some A − P, then in the witness sequence, we can move all of the )'s to the front to get ) ) 5 5 + œ +" ´ A# ´ â ´ A5" ´ A5 ´ A5" ´ â ´ A8" ´ +8 œ , and transitivity would imply that ) 5 + œ +" ´ A 5 ´ + 8 œ , Thus, in this case, the join takes the relatively simple form +Ð) ” 5Ñ, Í ) 5 +´B´, for some B − P. Definition Let ) and 5 be congruence relations on a lattice P. Define the product )5 by )5 +´, ) 5 if + ´ B ´ , for some B − P We say that ) and 5 commute if )5 œ 5) Thus, we are led to examine conditions under which congruence relations commute. The following facts are helpful in this regard. Lemma 9.7 Let ) and 5 be congruence relations on a lattice P. 1) If + Ÿ , then ) 5 + ´ B ´ , for some B − P Í ) 5 + ´ C ´ , for some C − Ò+ß ,Ó 188 Lattices and Ordered Sets 2) If +ß , Ÿ ?, then ) 5 +´?´, 5 Ê ) +´+•, ´, Proof. For part 1), since + Ÿ , , taking the join with + and the meet with , gives ) 5 +´B´, ) Ê 5 + ´ ÐB ” +Ñ • , ´ , For part 2), taking the meet with + gives +5Ð+ • ,Ñ and taking the meet with , gives Ð+ • ,Ñ), and so +5Ð+ • ,Ñ), . Theorem 9.8 Let ) and 5 be congruence relations on a lattice P. The following are equivalent: 1) ) and 5 commute. 2) For all + , , )5 +´, )5 Í +´, 3) )5 is a symmetric binary relation. 4) )5 œ ) ” 5. Proof. It is clear that 1) implies 2). To see that 2) implies 1), if +)5, then +)B5, for some B − P and so ) 5 + ´ Ð+ ” BÑ ´ Ð+ ” B ” ,Ñ Hence, Lemma 9.7 implies that 5 ) + ´ C ´ Ð+ ” B ” ,Ñ where + Ÿ C Ÿ + ” B ” , . Also, the symmetry of both ) and 5 imply that , 5B)+ and so a similar argument gives ) 5 , ´ D ´ Ð+ ” B ” ,Ñ where , Ÿ D Ÿ + ” B ” , . Thus, 5 ) 5 ) + ´ C ´ Ð+ ” B ” ,Ñ ´ D ´ , Since C and D are less than or equal to + ” B ” , , Lemma 9.7 implies that 5 5 ) ) + ´ C ´ ÐC • DÑ ´ D ´ , and so 5 ) +´C•D ´, whence +5), . This proves that +)5, implies +5), . The reverse implication follows by symmetry and so 1) and 2) are equivalent. Congruence Relations on Lattices 189 If ) and 5 commute, then )5 +´, 5) Í +´, Í )5 ,´, and so )5 is symmetric. Conversely, if )5 is symmetric, then 5) +´, )5 Í ,´+ Í )5 +´, and so ) and 5 commute. Thus, 1) and 3) are equivalent. Finally, we have seen that if ) and 5 commute, then ) ” 5 œ )5 and if ) ” 5 œ )5 , then )5 is symmetric. In 1950, Dilworth proved the following theorem. Theorem 9.9 (Dilworth [17]) Any two congruence relations on a relatively complemented lattice commute. Proof. Theorem 9.8 implies that it is sufficient to show that for any congruence relations ) and 5, )5 +´, 5) Ê +´, for all + , in P. If +)5, , then +)B5, , where B − Ò+ß ,Ó. Hence, as shown in Figure 9.3, b σ θ x x' θ σ a Figure 9.3 w if B is a relative complement of B in Ò+ß ,Ó, the quadrilateral property applied to Ð+ß Bß Bw ß ,Ñ gives +5Bw ), and so +5), . Quotient Lattices and Kernels It is now time to take a closer look at the properties of quotient lattices PÎ), where ) is a congruence relation on P. The set PÎ) of congruence classes of a congruence relation ) is also a lattice. Indeed, it is precisely because ) is a congruence relation that the inherited lattice operations of P are well defined on PÎ). Theorem 9.10 Let ) be a congruence relation on a lattice P. The set PÎ) of congruence classes is a lattice, called the quotient lattice of P modulo ) , under 190 Lattices and Ordered Sets the operations Ò+Ó • Ò,Ó œ Ò+ • ,Ó and Ò+Ó ” Ò,Ó œ Ò+ ” ,Ó Proof. First, we must check that these operations are well defined. But if + ´ +w and , ´ , w , then + • , ´ +w • , w and so Ò+Ó œ Ò+w Ó and Ò,Ó œ Ò, w Ó Ê Ò+ • ,Ó œ Ò+w • , w Ó which says that the meet operation is well defined. A similar argument applies to the join operation. Finally, it is routine to check the properties of a lattice. For instance, the associativity law ÐÒ+Ó ” Ò,ÓÑ ” Ò-Ó œ Ò+Ó ” ÐÒ,Ó ” Ò-ÓÑ is equivalent to ÒÐ+ ” ,Ñ ” -Ó œ Ò+ ” Ð, ” -ÑÓ which follows from associativity in P. We now describe the order relation in the quotient lattice. Theorem 9.11 Let P be a lattice and let ) − ConÐPÑ. 1) Ò+Ó Ÿ Ò,Ó if and only if some element of Ò+Ó is less than or equal to some element of Ò,Ó. In particular, +Ÿ, Ê Ò+Ó Ÿ Ò,Ó 2) Ò+Ó Ò,Ó if and only if α" or α ² " for every α − Ò+Ó and " − Ò,Ó, with strict inequality for at least one pair Ðα ß " Ñ . 3) Ò+Ó ² Ò,Ó if and only if every element of Ò+Ó is parallel to every element of Ò,Ó. 4) A congruence class ÒBÓ is an ideal of P if and only if it is the smallest element of PÎ), in which case it is called the kernel or ideal kernel of ) and is denoted by kerÐ)Ñ. If P has a !, then kerÐ)Ñ œ Ò!Ó. Proof. For part 1), + Ÿ , Í + • , œ + Ê Ò+ • ,Ó œ Ò+Ó Í Ò+Ó • Ò,Ó œ Ò+Ó Í Ò+Ó Ÿ Ò,Ó Conversely, Ò+Ó Ÿ Ò,Ó Ê Ò+ • ,Ó œ Ò+Ó Ê + • , − Ò+Ó and so α Ÿ " , where α œ + • , − Ò+Ó and " œ , − Ò,Ó. For part 2), if Ò+Ó Ò,Ó, then Ò+Ó Ÿ Ò,Ó, Ò+Ó Á Ò,Ó and Ò,Ó Ÿ y Ò+Ó. Thus, part 1) implies that α " for some Ðαß " Ñ − Ò+Ó ‚ Ò,Ó and that " Ÿ α cannot happen Congruence Relations on Lattices 191 for any Ðαß " Ñ − Ò+Ó ‚ Ò,Ó. Conversely, suppose that α " or α ² " for every Ðαß " Ñ − Ò+Ó ‚ Ò,Ó, with strict inequality for at least one pair. Then part 1) implies that Ò+Ó Ÿ Ò,Ó. However, Ò+Ó œ Ò,Ó implies that Ð+ß +Ñ − Ò+Ó ‚ Ò,Ó, which is false. Hence, Ò+Ó Ò,Ó. Part 3) follows from part 1). For part 4), let B − P. Since ÒBÓ is a sublattice, it is an ideal if and only if it is a down-set. If ÒBÓ is a down-set, then for any congruence class Ò+Ó, we have B • + − ÒBÓ and B • + Ÿ + − Ò+Ó and so ÒBÓ Ÿ Ò+Ó. Thus, ÒBÓ is the smallest element of PÎ)Þ Conversely, if ÒBÓ is the smallest element of PÎ) , then for any + Ÿ C − ÒBÓ we have Ò+Ó Ÿ ÒCÓ œ ÒBÓ and so Ò+Ó œ ÒBÓ, that is, + − ÒBÓ, which shows that ÒBÓ is a down-set. Figure 9.4 demonstrates that we may have ÒBÓ ÒCÓ but B ² C. 1 x [1] y [0] 0 Figure 9.4 Congruence Relations and Lattice Homomorphisms Now we want to look at the relationship between congruence relations and lattice homomorphisms. Theorem 9.12 1) Every lattice homomorphism 0 À P Ä Q defines a congruence relation )0 given by +)0 , Í 0 Ð+Ñ œ 0 Ð,Ñ and called the congruence kernel of 0 . The congruence classes of )0 are the sets PÎ)0 œ Ö0 " ÐBÑ ± B − imÐ0 Ñ× Thus, 0 is injective if and only if )0 is equality. 2) Every congruence relation ) on a lattice P defines a lattice epimorphism 1) À P Ä PÎ) given by 1) Ð+Ñ œ Ò+Ó and called the natural projection or canonical projection of P modulo ). 192 Lattices and Ordered Sets The congruence kernel of 1) is ) , that is, )1) œ ) Proof. For part 1), if +)0 , and B)0 C, then 0 Ð+Ñ œ 0 Ð,Ñ and 0 ÐBÑ œ 0 ÐCÑ and so 0 Ð+ ” BÑ œ 0 Ð+Ñ ” 0 ÐBÑ œ 0 Ð,Ñ ” 0 ÐCÑ œ 0 Ð, ” CÑ which implies that Ð+ ” BÑ)0 Ð, ” CÑ. A similar argument can be made for meets. For part 2), it is clear that 1) is surjective. Also, 1) Ð+ • ,Ñ œ Ò+ • ,Ó œ Ò+Ó • Ò,Ó œ 1) Ð+Ñ • 1) Ð,Ñ and similarly for join. Hence, 1) is a lattice epimorphism. Finally, the congruence kernel of 1) is ), since +) , Í Ò+Ó) œ Ò,Ó) Í 1) Ð+Ñ œ 1) Ð,Ñ Kernels Earlier in the book, we briefly discussed the ideal kernel of a lattice homomorphism. Let us repeat the definitions of the two types of kernels of a lattice homomorphism. Definition Let 0 À P Ä Q be a lattice homomorphism. 1) The congruence kernel of 0 is the congruence relation )0 . 2) If Q has a smallest element ! and if 0 " Ð!Ñ is nonempty, then the set kerÐ0 Ñ œ 0 " Ð!Ñ is called the ideal kernel of 0 . The ideal kernel of a homomorphism is easily seen to be an ideal. Also, as shown in Figure 9.5, the ideal kernel of the congruence kernel )0 is the ideal kernel of 0 . f congruence kernel ideal kernel θf ideal kernel ker(f)=f-1(0) Figure 9.5 Theorem 9.13 Let Q be a lattice with ! and let 0 À P Ä Q be a lattice homomorphism. Then the ideal kernel kerÐ0 Ñ œ 0 " Ð!Ñ is an ideal of P and kerÐ)0 Ñ œ kerÐ0 Ñ Congruence Relations on Lattices 193 Proof. We leave proof of part 1) for the reader. For part 2), it is easy to see that 0 " Ð!Ñ is an ideal of P and so Theorem 9.11 implies that kerÐ)0 Ñ œ 0 " Ð!Ñ. Universality and the Lattice Homomorphism Theorem The quotient lattice and its natural projection have a universal property. Theorem 9.14 Let ) be a congruence relation on a lattice P. The pair ÐPÎ)ß 1) Ñ has the following universal property of quotients: Referring to Figure 9.6, πθ L L/θ τ f M Figure 9.6 if 0 À P Ä Q is any lattice homomorphism for which ) © )0 , that is, ) +´, Ê 0 Ð+Ñ œ 0 Ð,Ñ then there is a unique lattice homomorphism 7 À PÎ) Ä M, which we call the mediating morphism for 0 , with the property that 7 ‰ 1) œ 0 The pair ÐPÎ)ß 1) Ñ is said to be universal. Also, 7 and 0 have the same image: imÐ7 Ñ œ imÐ0 Ñ and the same congruence kernel modulo ), in the sense that ÒBÓ)7 ÒCÓ Í B)0 C Proof. The condition 7 ‰ 1) œ 0 uniquely determines 7 to be 7ÐÒ:ÓÑ œ 0 Ð:Ñ This function is well defined on PÎ) since 0 is constant on Ò:Ó. Also, 7 ÐÒ:Ó • Ò;ÓÑ œ 7 ÐÒ: • ;ÓÑ œ 0 Ð: • ;Ñ œ 0 Ð:Ñ • 0 Ð;Ñ œ 7 ÐÒ:ÓÑ • 7 ÐÒ;ÓÑ and similarly for join. Hence, 7 is a lattice homomorphism. Finally, 194 Lattices and Ordered Sets imÐ7 Ñ œ Ö7 ÐÒ:ÓÑ ± : − P× œ Ö0 Ð:Ñ ± : − P× œ imÐ0 Ñ and ÒBÓ)7 ÒCÓ Í 7 ÒBÓ œ 7 ÒCÓ Í 0 ÐBÑ œ 0 ÐCÑ Í B)0 C Theorem 9.14 has a familiar corollary, which is obtained by taking ) œ )0 . Corollary 9.15 (The homomorphism theorem or first isomorphism theorem) Let 0 À P Ä Q be a lattice homomorphism. Then there is a unique lattice embedding 7 À PÎ)0 ä Q for which 7 ‰ 1)0 Ð:Ñ œ 7 ÐÒ:ÓÑ œ 0 Ð:Ñ Thus, 7 À PÎ)0 ¸ imÐ0 Ñ Proof. To see that 7 is injective, we have ÒBÓ)7 ÒCÓ Í B)0 C Í BO10 C Í ÒBÓ œ ÒCÓ There is also a correspondence theorem for lattices. Theorem 9.16 (The correspondence theorem) Let P be a lattice and let ) − ConÐPÑ. Then there is a bijection between the congruence relations on PÎ) and the congruence relations of P that contain ) . Proof. If 7 is a congruence on PÎ), then B7 w C if ÒBÓ) 7 ÒCÓ) is a congruence on P containing ) and if 5 is a congruence on P containing ) , then ÒBÓ) 5w ÒCÓ) if B5C is a congruence on PÎ). We leave the details to the reader. Subdirect Product Representations Definition Let P be a lattice and let Y œ ÖO3 ± 3 − M× be a nonempty family of lattices. An embedding 5À P Ä $3−M O3 is called a subdirect representation of P in #O3 if the projection maps 13 À P Ä O3 are surjective for all 3 − M , that is, if for any 5 − O3 , there is an + − P for which Ð5+ÑÐ3Ñ œ 5 . Theorem 9.17 Let P be a lattice. Congruence Relations on Lattices 195 1) Let Y œ Ö)3 ± 3 − M× be a nonempty family of congruence relations on P. If - œ 3)3 , then PÎ- has a subdirect representation in #PÎ)3 . In particular, if 3)3 is equality, then P has a subdirect representation in #PÎ)3 . 2) Conversely, let Y œ ÖO3 ± 3 − M× be a nonempty family of lattices. If P has a subdirect representation in #O3 , with projection maps 13 À P Ä O3 , then O3 ¸ PÎkerÐ13 Ñ and 3kerÐ13 Ñ is equality. Proof. For part 1), consider the map 5À PÎ- Ä #PÎ)3 defined by Ð5Ò+Ó- ÑÐ3Ñ œ Ò+Ó)3 which is well defined since +-+w implies that +)3 +w for all 3 − M . This is easily seen to be a homomorphism. Moreover, 5Ò+Ó- œ 5Ò,Ó- if and only if Ò+Ó)3 Ò,Ó for all 3 − M , which holds if and only if +3)3 , , that is, if and only if +-, . Hence, 5 is an embedding. It is clear that 13 is surjective for all 3. For part 2), since the projection maps are surjective, the first isomorphism theorem gives O3 ¸ PÎkerÐ13 Ñ and since +kerÐ13 Ñ, for all 3 if and only if 13 + œ 13 , for all 3, which holds if and only if + œ , , we see that 3kerÐ13 Ñ is equality. Standard Ideals and Standard Congruence Relations Our goal now is to describe conditions on a lattice P under which there is a bijection (in fact, a lattice isomorphism) between congruence relations on P and ideals of P. To this end, we define a special type of lattice element. Definition Let P be a lattice. An element 3 − P is standard if + • Ð3 ” ,Ñ œ Ð+ • 3Ñ ” Ð+ • ,Ñ for all +ß , − P. Note that standard elements might have been called join-standard elements, since there is a dual notion for this concept. Our primary interest is in standard elements of the ideal lattice \ ÐPÑ of a lattice P. Here is the definition in this context. For convenience, we introduce the new term friendly. Definition Let \ ÐPÑ be the ideal lattice of a lattice P. 1) An ideal M of P is standard if E • ÐM ” FÑ œ ÐE • MÑ ” ÐE • FÑ for all Eß F − \ ÐPÑ. Let f\ ÐPÑ denote the set of all standard ideals of P. 2) An ideal M of P is friendly if M ” E œ Ö3 ” + ± 3 − Mß + − E× for all E − \ ÐPÑ. Let Y \ ÐPÑ denote the set of all friendly ideals of P. 196 Lattices and Ordered Sets Theorem 9.18 An ideal M of a lattice P is friendly if and only if for any 3 − M and +ß , − P, , Ÿ+”3 Ê , œ Ð, • +Ñ ” 4 for some 4 − M . Proof. Suppose that M is friendly. If , Ÿ + ” 3 for 3 − M , then , − Æ + ” M and so , œ +w ” 4 for +w Ÿ + and 4 − M . But +w Ÿ , implies that , œ Ð+w • ,Ñ ” 4 Ÿ Ð+ • ,Ñ ” 4 Ÿ , and so , œ Ð+ • ,Ñ ” 4. The converse is clear. We next show that an ideal is standard if and only if it is friendly. Theorem 9.19 Let M be an ideal of a lattice P. The following are equivalent: 1) M is standard 2) M is standard with respect to principal ideals, that is, for all +ß , − P, Æ + • ÐM ” Æ ,Ñ œ Ð Æ + • MÑ ” Ð Æ + • Æ ,Ñ 3) M is friendly Proof. It is clear that 1) implies 2). To see that 2) implies 3), if , Ÿ + ” 3 for some 3 − M , then , œ , • Ð+ ” 3Ñ − Æ , • Ð Æ + ” MÑ œ Æ Ð+ • ,Ñ ” Ð Æ , • MÑ and so , Ÿ Ð+ • ,Ñ ” 4 for some 4 − M , 4 Ÿ , . Hence, , œ Ð+ • ,Ñ ” 4. To show that 3) implies 1), it is sufficient to show that E • ÐM ” FÑ © ÐE • MÑ ” ÐE • FÑ since the reverse inclusion is clear. If B − E • ÐM ” FÑ, then B − E and B œ 3 ” , for 3 − M and , − F . Thus, 3 ” , − E and so 3ß , − E, whence B œ 3 ” , − ÐE • MÑ ” ÐE • FÑ The standard ideals form a sublattice of the ideal lattice \ ÐPÑ. Theorem 9.20 The set f\ ÐPÑ of standard ideals of a lattice P is a sublattice of \ ÐPÑ, that is, the intersection and join of standard ideals is standard. Proof. Let M and N be standard. As to join, E • ÒÐM ” N Ñ ” FÓ œ E • ÒM ” ÐN ” FÑÓ œ ÐE • MÑ ” ÒE • ÐN ” FÑÓ œ ÐE • MÑ ” ÒÐE • N Ñ ” ÐE • FÑÓ œ ÒE • ÐM ” N ÑÓ ” ÐE • FÑ and so M ” N is standard. As to intersection M ∩ N , suppose that Congruence Relations on Lattices 197 , Ÿ+”5 where 5 − M ∩ N . Then writing - œ + • , , we have , œ-”3œ-”4 where 3 − M and 4 − N and 3 Ÿ - ” 4 implies that 3 œ Ð- • 3Ñ ” 4w for 4w − N . But 4w Ÿ 3 − M and so 4w − M ∩ N . Hence, , œ - ” 3 œ - ” Ð- • 3Ñ ” 4w œ - ” 4w œ Ð+ • ,Ñ ” 4w Hence, M ∩ N is standard. Standard Congruence Relations We can now describe the relationship between standard ideals and certain types of congruence relations. Definition Let P be a lattice and let M be an ideal of P. If the binary relation defined by B5M C if ÐB • CÑ ” 3 œ B ” C, for some 3 − M is a congruence relation on P, we say that 5M is standard. Note that if B Ÿ C, then B5M C if and only if there is an 3 − M for which B ” 3 œ C. Theorem 9.21 Let M − \ ÐPÑ. Then 5M is standard if and only if M is standard. Moreover, if M is standard, then a) 5M is the smallest congruence relation on P for which all elements of M are congruent b) kerÐ5M Ñ œ M . We denote the set of all standard congruence relations on P by StConÐPÑ. Proof. If 5 œ 5M is standard and if , Ÿ + ” 3, for 3 − M , then 5 , œ , • Ð+ ” 3Ñ ´ , • + and so there exists 4 − M for which , œ Ð, • +Ñ ” 4 Hence, M is standard. Conversely, suppose that M is standard. We show that 5 is a congruence relation on P using Theorem 9.4. It is clear that 5 is reflexive and it is easy to see that B5C if and only if ÐB • CÑ5ÐB ” CÑ. Next, suppose that B Ÿ C Ÿ D and B5C5D . Then there exist 3ß 4 − M for which B”3œC and C ” 4 œ D 198 Lattices and Ordered Sets and so B”3”4œC”4œD Finally, suppose that B Ÿ C, B5C and > − P. Then B ” 3 œ C for some 3 − M and so ÐB ” >Ñ ” 3 œ C ” > For the meet condition, since B ” 3 œ C, we have C • > Ÿ B ” 3 and so the standardness of M implies that C • > œ ÐC • > • BÑ ” 4 œ ÐB • >Ñ ” 4 for some 4 − M . Hence, ÐB • >Ñ5ÐC • >Ñ. For part a), it is easy to see that all elements of M are congruent. Moreover, if a congruence relation ) has the property that all elements of M are congruent and if B5M C, then ÐB • CÑ ” 3 œ B ” C for some 3 − M and so B • C œ ÒÐB • CÑ ” ÐB • C • 3ÑÓ)ÒÐB • CÑ ” 3Ó œ B ” C Hence, ÐB • CÑ)ÐB ” CÑ, which implies that B)C. Thus, 5M © ) . For part b), if 3 − M and B5M 3, then ÐB • 3Ñ ” 4 œ B ” 3 for some 4 − M and so B ” 3 − M , whence B − M . Thus, M is a congruence class and so kerÐ5M Ñ œ M . We have seen that f\ ÐPÑ is a sublattice of \ ÐPÑ. It is also true that StConÐPÑ is a sublattice of ConÐPÑ. Theorem 9.22 Let P be a lattice. 1) StConÐPÑ is a sublattice of ConÐPÑ. 2) For all Mß N − \ ÐPÑ, 5M∩N œ 5M ∩ 54 and 5M”N œ 5M ” 54 3) The maps 5À f\ ÐPÑ Ä StConÐPÑ and kerÀ StConÐPÑ Ä f\ ÐPÑ where 5ÐMÑ œ 5M are inverse lattice isomorphisms. Proof. Since kerÐ5M Ñ œ M , it follows that 5 is bijective, with inverse equal to the kernel map. For the intersection, it is clear that 5M∩N © 5M ∩ 5N and so it is sufficient to show that BÐ5M ∩ 5N ÑC implies B5M∩N C , where B C. In this case, there exist 3 − M and 4 − N for which B”3œC and B ” 4 œ C Hence, 3 Ÿ B ” 4 and since N is standard, there is a 4w − N for which Congruence Relations on Lattices 199 3 œ Ð3 • BÑ ” 4w But 4w Ÿ 3 implies that 4w − M and so C œ B ” 3 œ B ” 4w where 4w − M ∩ N . Hence, B5M∩N C and so 5M∩N œ 5M ∩ 54 . For the join, it is clear that 5M ß 5N © 5M”N and so 5M ” 5N © 5M”N . To see that B5M”N C implies BÐ5M ” 5N ÑC, again we assume that B C. Then there is a 5 − M ” N for which B ” 5 œ C. Since M is friendly, 5 œ 3 ” 4 for some 3 − M and 4 − N and so C œ B ” Ð3 ” 4Ñ. Therefore, 5N 5M B´B”3´B”3”4œC which shows that BÐ5M ” 5N ÑC. It follows that 5 is a lattice isomorphism, with inverse map ker. It is also true that standard congruence relations commute. Theorem 9.23 Any two standard congruence relations on a lattice P commute, that is, 5 M 5N œ 5N 5M for all standard ideals M and N . Proof. Let 5M and 5N be standard congruence relations on P. Theorem 9.8 implies that it is sufficient to show that + 5M 5N , Ê + 5N 5M , for all + , in P. But if + 5M B 5N , for some B − Ò+ß ,Ó, then there are 3 − M and 4 − N for which +”3œB and B”4œ, Hence, 5N 5M + ´ Ð+ ” 4Ñ ´ Ð+ ” 4 ” 3Ñ œ , and so +5N 5M , . Standard Ideals in Distributive Lattices We have seen that the maps 5À f\ ÐPÑ Ä StConÐPÑ and kerÀ StConÐPÑ Ä f\ ÐPÑ where 5ÐMÑ œ 5M are inverse lattice isomorphisms. Therefore, it is of obvious 200 Lattices and Ordered Sets interest to know when f\ ÐPÑ œ \ ÐPÑ and StConÐPÑ œ ConÐPÑ If P is a distributive lattice, then it is easy to see that all ideals of P are friendly, for if , Ÿ + ” 3 for 3 − M , then , œ , • Ð+ ” 3Ñ œ Ð, • +Ñ ” Ð, • 3Ñ where , • 3 − M . Hence, M is friendly. Moreover, if every ideal M is friendly, then every ideal M is the kernel of a congruence relation, namely, the standard congruence 5M . Finally, if every ideal M is the kernel of a congruence relation, then P is distributive, for if P contains one of the forbidden sublattices in Figure 9.7, b x b y x z y z a M3 a N5 Figure 9.7 then the ideal Æ D is not a kernel. To see this, if Æ D œ kerÐ)Ñ, then +)D and so the quadrilateral property implies that , )C, which in turn implies that +)B and so B − Æ D, which is false. Theorem 9.24 The following are equivalent for a lattice P: 1) P is distributive. 2) Every ideal of P is friendly (standard), that is, f\ ÐPÑ œ \ ÐPÑ 3) Every ideal of P is the kernel of some congruence relation on P. As to the issue of when StConÐPÑ œ ConÐPÑ, note that the definition of a standard congruence is + 5M , if Ð+ • ,Ñ ” 3 œ + ” , for some 3 − M and this is one half of the requirements for 3 to be a relative complement of + • , with respect to the section Ò!ß + ” ,Ó. Congruence Relations on Lattices 201 Indeed, if P is sectionally complemented, then every congruence relation ) on P is standard; in fact, ) œ 5kerÐ)Ñ To see this, it is clear that B5M C Í ÐB • CÑ5M ÐB ” CÑ for any standard congruence 5M and since this also holds for ), it is sufficient to show that +), Í +5kerÐ)Ñ , for all + Ÿ , . If +), with + Ÿ , and if C is a relative complement of + in Ò!ß ,Ó, then +”C œ, and + • C œ ! But +), implies that Ð+ • CÑ)Ð, • CÑ, that is, !)C . Hence C − kerÐ)Ñ and so +5kerÐ)Ñ , . Conversely, if +5kerÐ)Ñ , with + Ÿ , , then there is a C − kerÐ)Ñ for which + ” C œ , . But +)Ð+ ” CÑ and so +), . Hence ) œ 5kerÐ)Ñ . Thus, if P is sectionally complemented, then all congruence relations on P are standard. In looking for a converse, suppose that all congruence relations on P are standard and let - − Ò!ß ,Ó. We wish to show that there is an 3 − P for which -”3œ, and -•3œ! If ) is any congruence relation on P, then ) œ 5M for some ideal M and so the condition - ” 3 œ , follows from - ), . Also, since kerÐ)Ñ œ M , if we can find a ) for which kerÐ)Ñ œ f Ð-Ñ œ ÖB − P ± - • B œ !× then we are done. Now, fÐ-Ñ is not an ideal in general, but it is an ideal if P is distributive. Moreover, it appears that one of the congruence relations of Example 9.3: B.C if B•- œC•- is just what we need, since - ., and kerÐ.Ñ œ ÖB − P ± B • - œ !× œ f Ð-Ñ We can now put the pieces together and add one more piece. Theorem 9.25 Let P be a lattice. The maps 5À f\ ÐPÑ Ä StConÐPÑ and kerÀ StConÐPÑ Ä f\ ÐPÑ where 5ÐMÑ œ 5M are inverse lattice isomorphisms. Moreover: 1) P is distributive if and only if \ ÐPÑ œ f\ ÐPÑ. 2) If P is sectionally complemented, then ConÐPÑ œ StConÐPÑ. 3) The following are equivalent: a) P is distributive and sectionally complemented 202 Lattices and Ordered Sets b) \ ÐPÑ œ f\ ÐPÑ and ConÐPÑ œ StConÐPÑ c) The maps 5À \ ÐPÑ Ä ConÐPÑ and kerÀ ConÐPÑ Ä \ ÐPÑ are inverse bijections, in which case these maps are inverse lattice isomorphisms. d) P is distributive and all congruence relations on P commute. Proof. We have seen that 3a) and 3b) are equivalent and it is clear that 3b) implies 3c). If 3c) holds, then all ideals M are standard, since 5ÐMÑ œ 5M is a congruence relation. Also, all congruence relations ) are standard, since ) œ 5ÐMÑ œ 5M , for some ideal M . Hence, 3c) implies 3b). If 3a) holds, then every section Ò!ß +Ó of P is modular and complemented and therefore relatively complemented. Hence, P is relatively complemented and so Theorem 9.9 implies that all congruence relations on P commute. Finally, assume that 3d) holds. We show that P is sectionally complemented. Let > − Ò!ß ,Ó and consider the congruence relations +$ , if +”>œ,”> + ., if +•>œ,•> and of Example 9.3. Then !$ >., implies that !.B$ , , for some B − Ò!ß ,Ó, that is, !œ>•B and B ” > œ , Hence, B is a relative complement of > in Ò!ß ,Ó and P is sectionally complemented. Exercises 1. 2. 3. 4. 5. Prove that a subset ) of P# is a congruence relation on P if and only if the following hold: a) ) is an equivalence relation on P. b) ) is a sublattice of the lattice P# (under coordinatewise meet and join). What is the smallest element of ConÐPÑ? What is the largest element? Let P be a lattice and let ) − ConÐPÑ. Prove that Ò+Ó ö Ò,Ó if and only if a) Ò+Ó Ò,Ó and b) + Ÿ B Ÿ , Ê B − Ò+Ó or B − Ò,Ó. Shows that, in general, even when P is distributive, congruence kernels need not be distinct, that is, there can be two distinct congruence relations on P that have the same kernel. A lattice P is simple if ConÐPÑ œ Ö!ß "× where ! is the smallest congruence and " is the largest congruence. Show that the lattice Q8 shown in Figure 9.8 is simple for 8 $. Congruence Relations on Lattices 203 1 x1 x2 xn 0 Mn Figure 9.8 6. Let F be a Boolean lattice. Prove that a congruence relation ) on the lattice F is also a Boolean congruence, that is, +), implies +- ), - . 7. Let 0 À P Ä Q be a lattice homomorphism where P and Q have !. Characterize the fact that 0 is a lattice embedding in terms of the congruence kernel )0 . What about the ideal kernel of 0 (if it exists)? 8. If P is a distributive (modular) lattice, show that PÎ) is also distributive (modular). 9. Let M be a standard ideal in a lattice P. Prove that the standard congruence relation 5M is the smallest congruence relation on P for which all elements of M are congruent. 10. Let ) and 5 be congruence relations on a lattice P. Show that if ) Ÿ 5 then there is a unique epimorphism 0 À PÎ) Ä PÎ5 such that 0 ‰ 1) œ 15 . 11. Two lattice epimorphisms 0 ß 1 − EpiÐPÑ are kernel equivalent if their congruence kernels )0 and )1 are equal, that is, if 0 Ð+Ñ œ 0 Ð,Ñ Í 1Ð+Ñ œ 1Ð,Ñ a) Show that this is an equivalence relation ´ . b) Prove that the lattice epimorphisms 0 À P Ä Q and 1À P Ä R are kernel equivalent if and only if there is a lattice isomorphism 5À Q Ä R for which 5 ‰ 0 œ 1. c) Let P be a bounded lattice. Show that the map ) È Ò1) Ó sending ) − ConÐPÑ to the equivalence class containing 1) is bijective and so ConÐPÑ Ç EpiÐPÑÎ ´ where EpiÐPÑ is the set of all epimorphisms with domain P. 12. Prove the following theorem, which Grätzer [24] refers to as the first isomorphism theorem for lattices. Let ) be a congruence relation on a lattice P. Let W be a set of canonical forms for ), that is, W is a set consisting of exactly one member from each congruence class of ) . If W is a sublattice of P then W ¸ PÎ) . 13. Let Ö)3 ± 3 − M× be a family of congruence relations on a lattice P. Prove that if all of these congruences have kernels then kerÐ4)3 Ñ œ ,kerÐ)3 Ñ 204 Lattices and Ordered Sets Thus, the family ^ of kernels inherits meets from \ ÐPÑ and so if P has ! then ^ is a complete lattice. 14. Let P be a lattice and let M ü P. a) Prove that B ” 3 œ C ” 3 for some 3 Í ÆB ” M œ ÆC ” M b) Let 3 − M . Prove that the following are equivalent: i) 3 ” ÐB • CÑ œ Ð3 ” BÑ • Ð3 ” CÑ for all Bß C − P. ii) The join map 43 À B È B ” 3 is a homomorphism. iii) The binary relation defined by B)3 C if B ” 3 œ C ” 3 is a congruence relation. 15. Let P be a lattice and let +ß , − P. The smallest congruence relation )+ß, on P for which +)+ß, , is called the principal congruence generated by the pair Ð+ß ,Ñ. a) Show that )+ß, œ 4Ö5 − ConÐPÑ ± +5,× b) Show that for any 7 − ConÐPÑ, 7 œ 2Ö)+ß, ± +7 ,× Hence, the family of principal congruence relations is join-dense in ConÐPÑ. c) Prove that the principal congruence relations are compact and so ConÐPÑ is algebraic. d) Find the principal congruence ) Bß" for the lattice R& in Figure 9.9. 1 x y z 0 N5 Figure 9.9 16. a) Prove that a nonempty subset W of a lattice P is a convex sublattice of P if and only if W œ M ∩ J , where M is an ideal and J is a filter. b) Let P be a lattice and let ) be a congruence relation on P. A filter J in P is called the cokernel of ) if J is the largest element in the quotient PÎ). Let M be a kernel in P and let J be a cokernel in P. Show that if M ∩ J is nonempty, then it is a congruence class for some congruence relation on P. Congruence Relations on Lattices 205 c) Prove that a lattice P is distributive if and only if every convex sublattice of P is a congruence class. 17. Let P be a distributive lattice and let M ü P and J ý P. Define a binary relation ) on P by B) C a) b) c) d) e) f) if ÐB ” ?Ñ • @ œ ÐC ” ?Ñ • @ for some ? − M and @ − J Prove that ) is a congruence relation on P. If P is bounded, prove that M © Ò!Ó. If P is bounded, prove that J © Ò"Ó. If P is bounded, prove that ) is the smallest congruence relation on P for which M © Ò!Ó and J © Ò"Ó. Prove that ) is proper () Á P ‚ P) if and only if M ∩ J œ g. Prove that ) is trivial (equality) if and only if M œ Ö!× and J œ Ö"×. Distributive Ideals Definition Let P be a lattice. 1) An element 3 − P is distributive if 3 ” Ð+ • ,Ñ œ Ð3 ” +Ñ • Ð3 ” ,Ñ for +ß , − P. 2) An ideal M − \ ÐPÑ is distributive if M ” ÐE • FÑ œ ÐM ” EÑ • ÐM ” FÑ for Eß F − \ ÐPÑ. Let W\ ÐPÑ denote the set of all distributive ideals of P. 3) If the binary relation defined by B$M C if B ” 3 œ C ” 3, for some 3 − M is a congruence relation on P, we say that $M is distributive. 18. Let M be an ideal of a lattice P. Prove that M is standard if and only if both of the following hold: a) M is distributive b) Relative complements of M , if they exist, are unique, that is, for all Eß F − \ ÐPÑ, M •E œM •F and M ” E œ M ” F Ê EœF Find a distributive ideal that is not standard. Hint: Look in a small nondistributive lattice. 19. Show that for a distributive congruence relation B$M C Í ÆB ” M œ ÆC ” M 20. Let M be a distributive ideal in a lattice P. Prove that the distributive congruence relation $M is the smallest congruence relation on P for which all elements of M are congruent. 206 Lattices and Ordered Sets 21. Let M be an ideal of a lattice P. Prove that $M is distributive if and only if M is distributive. Moreover, if M is distributive, then a) $M is the smallest congruence relation on P for which all elements of M are congruent b) kerÐ$M Ñ œ M . Prove that if M is a standard ideal, then 5M œ $M . 22. Find an example of a distributive congruence that is not standard. 23. Prove that if every ideal of a lattice P is distributive, then every ideal is standard. 24. Let P be a sectionally complemented lattice with the ACC. Prove that for any congruence relation ) on P there is an element 7 − P such that +) , for all +ß , − P. Í +”7œ,”7 Part II: Topics Chapter 10 Duality for Distributive Lattices: The Priestley Topology Let us repeat the representation theorem for distributive lattices (Theorem 6.6). Theorem 10.1 Let P be a nontrivial distributive lattice and let 3P À P Ä bÐSpecÐPÑÑ be the rho map for P, defined by 3P Ð+Ñ œ cc+ ³ ÖT − SpecÐPÑ ± +  T × 1) (Representation theorem for distributive lattices, Birkhoff, 1933 [6]) The rho map 3 is an embedding of P into the power set kÐSpecÐPÑÑ and so P is isomorphic to a power set sublattice (ring of sets). Moreover, if P is finite, then 3 is an isomorphism and P ¸ bÐSpecÐPÑÑ. 2) (Representation theorem for Boolean algebras) If P is a Boolean algebra, then 3 is a Boolean algebra embedding of P into the power set kÐSpecÐPÑÑ and so P is isomorphic to a power set subalgebra (field of sets). Moreover, if P is finite, then 3 is an isomorphism. There is more to the story concerning the representation of distributive lattices as power set sublattices. The family 3 œ Ö3P À P ¸ bÐSpecÐPÑÑ ± P is a finite distributive lattice× of lattice isomorphisms is part of a general duality between finite distributive lattices and finite posets. This duality also exists for arbitrary bounded distributive lattices, except that the rho maps are not surjective in this case and so we must replace the sets bÐSpecÐPÑÑ with other sets that are rather more difficult to describe. Indeed, in the early 1930s, Marshall Stone introduced the idea of topologizing the spectrum SpecÐFÑ of a Boolean algebra F to help describe the image of the S. Roman (ed.), Lattices and Ordered Sets, doi: 10.1007/978-0-387-78901-9_10, © Steven Roman 2008 209 210 Lattices and Ordered Sets rho maps. In particular, the sets cc+ form a basis for the Stone topology and are actually clopen (open and closed) in that topology. In the late 1930s, Stone generalized his topology to deal with bounded distributive lattices. Unfortunately, the resulting topology is not Hausdorff, although it is compact. Nonetheless, the introduction of topological methods led to a new connection between algebra and topology that has had a great influence in other areas of mathematics, including functional analysis, which was actually Stone's main area of interest. In the 1970s, Hilary Priestley [52] recognized that a different topology might be a more appropriate choice for the representation theory of arbitrary bounded distributive lattices. The Priestley topology on the spectrum SpecÐPÑ of a bounded distributive lattice has basis U œ Öcc+ß, ³ cc+ ∩ c, ± +ß , − P× This topology is both compact and Hausdorff. As we will see, it enables us to describe the image of the rho map in elegant terms as the family of all clopen down-sets in SpecÐPÑ. Our plan in this chapter is twofold. First, we will discuss the duality between finite distributive lattices and finite posets, since this does not require the introduction of topological notions. (More precisely, the Priestley topology is discrete when the sets are finite.) Then we will discuss the necessary topological notions so that we can present the general duality theory of bounded distributive lattices. Both dualities are most elegantly described using the language of category theory. Therefore, since some readers may not be familiar with this language, it seems appropriate to include an informal discussion here and, for those who desire a more formal approach, an appendix on the subject. We suggest that all readers review at least the informal discussion. An Informal Introduction to the Language of Category Theory We will need three main concepts from category theory: categories, functors and natural transformations. Categories Informally speaking, a category consists simply of a class of mathematical objects and their structure-preserving maps. Examples are the category of sets, the category of groups, the category of posets and the category of lattices. More specifically, a category V consists of a class ObjÐVÑ whose members are called the objects of the category. One often writes E − V as a shorthand for E − ObjÐVÑ. Also, for each pair ÐEß FÑ of objects, there is a set homV ÐEß FÑ, whose elements are called morphisms, maps or arrows. Often but not always, Duality 211 the morphisms in homV ÐEß FÑ are functions from E to F with special properties. The set homV ÐEß FÑ is called a hom set. It is common to denote the fact that 0 − homV ÐEß FÑ by writing 0 À E Ä F . The morphisms in a category must satisfy two conditions: There must be an associative composition and each object must have an identity morphism. Specifically, for 0 À E Ä F and 1À F Ä G , there is a morphism 1 ‰ 0 À E Ä G . Moreover, composition is associative: 0 ‰ Ð1 ‰ 2Ñ œ Ð0 ‰ 1Ñ ‰ 2 whenever the compositions are defined. Also, for each object E − ObjÐVÑ, there is a morphism +E À E Ä E called the identity morphism for E, with the property that if 0 À E Ä F , then +F ‰ 0 œ 0 and 0 ‰ +E œ 0 Finally, a morphism 0 À E Ä F is an isomorphism if there is a morphism 0 " À F Ä E for which 0 " ‰ 0 œ +E and 0 ‰ 0 " œ +F Example 10.2 In the category Set of sets, the class of objects is the class of all sets and the hom set homÐEß FÑ is the set of all functions from E to F . In the category Grp of groups, the class of objects is the class of all groups and the hom set homÐEß FÑ is the set of all group homomorphisms from E to F . In the category FinPoset of finite posets, the class of objects is the class of all finite posets and the morphisms from E to F are the order-preserving maps from E to F . In the category FinDLat of finite distributive lattices, the class of objects is the class of all finite distributive lattices and the morphisms from E to F are the lattice Ö!ß "×-homomorphisms from E to F . We give more examples of categories in the appendix. A subcategory W of a category V is a category with the property that every object of W is an object of V and every morphism of W is a morphism of V and that the composition and identities are the same in W as in V. Functors Structure-preserving “maps” between categories are called functors. Since a category consists of both objects and morphisms, a functor must consist of two functions—one to handle the objects and one to handle the morphisms. Also, there are two versions of functors: covariant and contravariant. More specifically, let V and W be categories. A functor J À V Ê W is a pair of functions (as is customary, we use the same symbol J for both functions): 212 Lattices and Ordered Sets 1) An object function J À ObjÐVÑ Ä ObjÐWÑ that maps objects in V to objects in W. 2) For a covariant functor, a morphism function J À MorÐVÑ Ä MorÐWÑ that maps each morphism 0 À E Ä F in V to a morphism J 0 À J E Ä J F in W. For a contravariant functor, a morphism function J À MorÐVÑ Ä MorÐWÑ that maps each morphism 0 À E Ä F in V to a morphism J 0 À J F Ä J E in W (note the reversal of direction). 3) Identity and composition are preserved, that is, for a covariant functor, and J Ð1 ‰ 0 Ñ œ J 1 ‰ J 0 J +E œ +J E and for a contravariant functor, J +E œ + J E and J Ð1 ‰ 0 Ñ œ J 0 ‰ J 1 whenever all compositions are defined. The notation J À V Ê W is read “J is a functor from V to W.” A functor J À V Ê V from V to itself is referred to as a functor on V . One way to think of a covariant functor J À V Ê W is as a mapping of one-arrow diagrams in V 0 EÒF to one-arrow diagrams in W J0 JE Ò JF with the property that identity and composition are preserved. Example 10.3 The power set functor cÀ Set Ê Set sends a set E to its power set kÐEÑ and each set function 0 À E Ä F to the induced function c0 À kÐEÑ Ä kÐFÑ that sends a subset \ © E to 0 \ . It is easy to see that this defines a covariant functor. More to the purpose of this book, the spectrum functor J À FinDLat Ê FinPoset sends a finite distributive lattice P to its spectrum SpecÐPÑ and a lattice homomorphism 0 À P Ä Q to the induced inverse map 0 " À SpecÐQ Ñ Ä SpecÐPÑ Duality 213 restricted to prime ideals. In the opposite direction, the down-set functor bÀ FinPoset Ê FinDLat sends a finite poset T to the lattice bÐT Ñ of down-sets of T and a monotone map 1À T Ä U between posets to the induced inverse map 1" À bÐUÑ Ä bÐT Ñ, restricted to down-sets. We will show later that these are contravariant functors. Functors can be composed in the “obvious” way. Specifically, if J À V Ê W and KÀ W Ê X are functors, then K ‰ J À V Ê X is defined by ÐK ‰ J ÑÐEÑ œ KÐJ EÑ and ÐK ‰ J ÑÐ0 Ñ œ KÐJ 0 Ñ for E − ObjÐVÑ and 0 − homV ÐEß FÑ. The morphism portion of a functor J À V Ê W maps hom sets to hom sets, specifically, J À homÐEß FÑ Ä homÐJ Eß J FÑ or J À homÐEß FÑ Ä homÐJ Fß J EÑ in the case of a contravariant functor. If each of these hom set maps is surjective, then J is said to be full and if each of these hom set maps is injective, then J is said to be faithful. Natural Transformations A structure-preserving map between functors is called a natural transformation. Consider a pair of covariant functors J ß KÀ V Ê W, as shown in Figure 10.1. FA F A f Ff λA B G FB λB GA Gf GB Figure 10.1 We have remarked that the essence of a functor is what it does to the one-arrow † K is intended to map, diagrams 0 À E Ä F . A natural transformation -À J Ä in a nice way, the family of one-arrow diagrams that characterize the functor J to the corresponding family of one-arrow diagrams that characterize K. As shown in Figure 10.1, this is accomplished by a family Ö-E À J E Ä KE ± E − ObjÐVÑ× 214 Lattices and Ordered Sets of morphisms in W—one for each object in V— with the property that the square in Figure 10.1 commutes, that is, K0 ‰ -E œ -F ‰ J 0 † K or Ö-E ×À J Ä † K to denote a natural transformation. Each We write -À J Ä map -E is called a component of the natural transformation. If each component -E is an isomorphism, then - is called a natural † " isomorphism and we write Ö-E ×À J ¸ K. In this case, the family Ö-E × is a natural isomorphism from K to J . Some very important commonly known maps are natural transformations. We give two examples in the appendix: the determinant and the coordinate map for finite-dimensional vector spaces. In this chapter, we will show that the family 3 œ Ö3P À P ¸ bÐSpecÐPÑÑ ± P is a finite distributive lattice× of rho maps is a natural isomorphism. Dual Categories It seems reasonable to say that two categories V and W are isomorphic if there exist functors J À V Ê W and KÀ W Ê V for which J ‰ K œ MW and K ‰ J œ MV where MV is the identity functor on V and similarly for W. However, this notion turns out to be too restrictive. A more useful notion comes when we require only that the two compositions J ‰ K and K ‰ J are naturally isomorphic to the respective identity functors. Thus, two categories V and W are naturally isomorphic (or naturally equivalent) if there are functors J À V Ê W and KÀ W Ê V for which † J ‰ K ¸ MW and † K ‰ J ¸ MV When the functors J and K are contravariant, this condition is often expressed by saying that V and W are dual categories. We will show that the categories FinDLat and FinPoset are dual, as are the categories BDLat of (arbitrary) bounded distributive lattices and OrdBool of ordered Boolean spaces (defined later). In fact, these dualities are the subject of this chapter. Finally, it is not hard to see that if J À V Ê W and KÀ W Ê V are functors for which Duality † J ‰ K ¸ MW and 215 † K ‰ J ¸ MV then J and K are both full and faithful. This follows from the fact that functors are ordinary functions when restricted to hom sets. The Duality Between Finite Distributive Lattices and Finite Posets The duality between finite distributive lattices and finite posets is described in Figure 10.2. f L M ρL S ρM O(Spec(L)) (f-1)-1 O(Spec(M)) O FinDLat g X Y κX κY Spec(O(X)) (g-1)-1 Spec(O(Y)) FinPoset Figure 10.2 The Categories and Functors Let FinDLat be the category of all finite distributive lattices, with lattice Ö!ß "×homomorphisms and let FinPoset be the category of all finite posets, with monotone functions. The spectrum functor fÀ FinDLat Ê FinPoset is defined by f ÐPÑ œ SpecÐPÑ and f Ð0 Ñ œ 0 " À SpecÐQ Ñ Ä SpecÐPÑ where P and Q are finite distributive lattices and 0 À P Ä Q is a Ö!ß "×homomorphism. Note that the map 0 " is the induced inverse map, restricted to the prime ideals of Q . It is easy to see that f is a contravariant functor, since f Ð+Ñ œ +" œ + and f Ð1 ‰ 0 Ñ œ Ð1 ‰ 0 Ñ" œ 0 " ‰ 1" œ f Ð0 Ñ ‰ f Ð1Ñ The down-set functor bÀ FinPoset Ê FinDLat is defined by bÀ T È bÐT Ñ and bÐ1Ñ œ 1" À bÐUÑ Ä bÐT Ñ where T and U are finite posets and 1À T Ä U is a monotone map. The map 216 Lattices and Ordered Sets 1" is the induced inverse map, restricted to the down-sets of U. It is clear that b is also a contravariant functor. The Natural Isomorphisms We have seen that the rho maps: 3 œ Ö3P À P ¸ bÐSpecÐPÑÑ ± P is a finite distributive lattice× are lattice isomorphisms. On the FinPoset side, we define a family of order isomorphisms , œ Ö,T À T ¸ SpecÐbÐT ÑÑ ± T is a finite poset× by ,T Ð+Ñ œ ^c+ ³ ÖH − bÐT Ñ ± +  H× It is easy to see that ^c+ is a prime ideal in bÐT Ñ. Since T is finite, all filters in bÐT Ñ are principal. Since bÐT Ñ is distributive, Theorem 4.26 implies that the prime filters are precisely the filters of the form Å bÐT Ñ ÐHÑ where H − bÐT Ñ is join-irreducible. Hence, the prime ideals are precisely the ideals of the form Ð Å bÐT Ñ ÐHÑÑ- where H − bÐT Ñ is join-irreducible. But Example 3.20 shows that the join-irreducibles in bÐT Ñ are the prime ideals Æ T +. Thus, the prime ideals in bÐT Ñ are precisely the ideals of the form ˆ Å bÐT Ñ Ð Æ T +щ- œ Ð^+ Ñ- œ ^c+ This shows that ,T is surjective. To see that ,T is an order isomorphism, we have +Ÿ, Ê ^c+ © ^c, and ^c+ © ^c, Ê Æ ,  ^c+ Ê + − Æ, Ê +Ÿ, The fact that the families 3 and , are natural isomorphisms is shown in the diagrams in Figure 10.2Þ Note that the functors f and b are defined in such a way that Ð0 " Ñ" À bÐSpecÐPÑÑ Ä bÐSpecÐQ ÑÑ and Ð1" Ñ" À SpecÐbÐT ÑÑ Ä SpecÐbÐUÑÑ Now, the diagram on the left in Figure 10.2 commutes if and only if 3Q ‰ 0 œ Ð0 " Ñ" ‰ 3P that is, for all + − P, Duality 217 cc0 Ð+Ñ œ Ð0 " Ñ" cc+ But this holds since Ð0 " Ñ" cc+ œ Ð0 " Ñ" ÐÖT − SpecÐPÑ ± +  T ×Ñ œ ÖU − SpecÐQ Ñ ± +  0 " ÐUÑ× œ ÖU − SpecÐQ Ñ ± 0 Ð+Ñ Â U× œ cc0 Ð+Ñ Proof that the second diagram commutes: ,U ‰ 1 œ Ð1" Ñ" ‰ ,T is similar. Thus, 3 and , are natural isomorphisms and the categories FinDLat and FinPoset are dual. Theorem 10.4 The categories FinDLat and FinPoset are dual. Specifically: 1) The following are contravariant functors: a) fÀ FinDLat Ê FinPoset defined by f À P È SpecÐPÑ and f Ð0 Ñ œ 0 " À SpecÐQ Ñ Ä SpecÐPÑ for 0 À P Ä Q . b) bÀ FinPoset Ê FinDLat defined by bÀ T È bÐT Ñ and bÐ1Ñ œ 1" À bÐUÑ Ä bÐT Ñ for 1À T Ä U. 2) The family of rho maps 3 œ Ö3P À P ¸ bÐSpecÐPÑÑ ± P − FinDLat× defined by 3P Ð+Ñ œ cc+ ³ ÖM − SpecÐPÑ ± +  M× is a natural isomorphism 3À MFinDLat Ä b ‰ f . The family of kappa maps , œ Ö,T À T ¸ SpecÐbÐT ÑÑ ± T − FinPoset× defined by ,T Ð:Ñ œ ^c: ³ ÖH − bÐT Ñ ± :  H× is a natural isomorphism ,À MFinPoset Ä f ‰ b . 3) The functors f and b are full and faithful. That is: a) f defines a bijection between hom sets: fÀ homFinDLat ÐPß Q Ñ Ç homFinPoset ÐSpecÐQ Ñß SpecÐPÑÑ 218 Lattices and Ordered Sets b) b defines a bijection between hom sets: bÀ homFinPoset ÐT ß UÑ Ç homFinDLat ÐbÐUÑß bÐT ÑÑ Proof. For part 3), see Theorem A2.8. Totally Order-Separated Spaces As mentioned earlier, in order to describe the duality theory for arbitrary bounded distributive lattices, we require a few basic notions from point-set topology, which are summarized in a short appendix. We also require a few more specialized facts, presented here. Proofs will, in general, be left to the reader. Definition An ordered topological space is a triple Ð\ß 7 ß Ÿ Ñ where Ð\ß 7 Ñ is a topological space and Ð\ß Ÿ Ñ is a poset. We will denote the set of all clopen sets in \ by k clo Ð\Ñ and the set of all clopen down-sets in \ by b clo Ð\Ñ. Definition If Ð\ß 7 ß Ÿ Ñ and Ð] ß 5ß Ÿ Ñ are ordered topological spaces then a map 0 À \ Ä ] is an order-homeomorphism if it is an order isomorphism of posets and a homeomorphism of topological spaces. Definition 1) A topological space Ð\ß 7 Ñ is totally separated if the clopen sets separate points, that is, if BÁC Ê bH − k clo Ð\Ñ with C − H and B  H A compact, totally separated topological space is called a Boolean space or Stone space. 2) An ordered topological space Ð\ß 7 ß Ÿ Ñ is totally order-separated if the clopen down-sets separate points, that is, if BŸ y C Ê bH − bclo Ð\Ñ with C − H and B  H A compact, totally order-separated ordered topological space is called an ordered Boolean space or a Priestley topological space. We note the following: 1) Totally separated and totally order-separated spaces are Hausdorff. 2) In a totally order-separated space, the order is determined by the clopen down sets, since BŸC Í ÐC − H Ê B − H for all H − bclo Ð\ÑÑ 3) Since a Boolean space or an ordered Boolean space \ is compact and Hausdorff, the closed subsets of \ are the same as the compact subsets of \. It follows that any clopen set is a finite union of basic open sets. Duality 219 4) Any Boolean space \ is an ordered Boolean space under the partial order of equality. For in this case, B Ÿ C means B œ C and so every nonempty subset of \ is a down-set. Since C Ÿ y B means C Á B and since \ is totally separated, there is a clopen (down-) set O for which B − O and C  O . Thus, \ is totally order-separated and so is an ordered Boolean space. Moreover, bclo Ð\Ñ œ k clo Ð\Ñ Conversely, if \ is an ordered Boolean space under equality, then bclo Ð\Ñ œ k clo Ð\Ñ and \ is a Boolean space. In a totally order-separated space, separation can be extended to certain compact subsets. Theorem 10.5 Let \ be totally order-separated. If E and F are disjoint compact sets with the property that no element of F is less than any element of E, that is, if F ∩ Ð Æ EÑ œ g then there is a clopen down-set G such that E © G and F ∩ G œ g. Of course, the intersection of clopen down-sets is a closed down-set and the union of clopen down-sets is an open down-set. In an ordered Boolean space, the converses are also true. Theorem 10.6 Let \ be an ordered Boolean space. 1) The closed down-sets of \ are precisely the intersections of clopen downsets. 2) The open down-sets of \ are precisely the unions of clopen down-sets. Theorem 10.7 If G is a closed subset of an ordered Boolean space \ , then Æ G and Å G are also closed. The Priestley Prime Ideal Space Let P be a bounded lattice. We gather a few facts about the sets cc+ , c, and cc+ß, ³ cc+ ∩ c, for +ß , − P. Theorem 10.8 Let P be a nontrivial bounded lattice and let f œ Öcc+ ± + − P× Then f © U since and U œ Öcc+ß, ± +ß , − P× 220 Lattices and Ordered Sets cc+ œ cc+ß! for all + − P. Moreover, 1) f is a sublattice of the power set lattice kÐSpecÐPÑÑ, in particular, cc+ ∩ cc, œ ccÐ+•,Ñ and cc+ ∪ cc, œ ccÐ+”,Ñ If P is complemented, then f is a subalgebra of kÐSpecÐPÑÑ, that is, c+ œ Ðcc+ Ñ- œ ccÐ+w Ñ 2) U has finite intersections, since cc+" ß," ∩ cc+# ß,# œ ccÐ+" •+# ÑßÐ," ”,# Ñ The complement of an element of U is cc+ß, œ cc,ß! ∪ cc"ß+ 3) If P is complemented, then c, œ cc,w and so cc+ß, œ ccÐ+•,w Ñ Thus, U œ f . Since f and U each have finite intersections, Theorem A1.2 implies that they can each serve as a basis for a topology on SpecÐPÑ. Moreover, since œ cc,ß! ∪ cc"ß+ cc+ß, each basis element cc+ß, is clopen in this topology. Definition Let P be a bounded distributive lattice. 1) The ordered topological space ÐSpecÐPÑß 7 ß © Ñ, where 7 is the topology with basis consisting of the clopen sets U œ Öcc+ß, ± +ß , − P× is called the Priestley prime ideal space or the Priestley dual space of P. Note that the sets Öcc+ ± + − P× ∪ Öc, ± , − P× form a subbasis for the Priestley prime ideal space. 2) The topological space ÐSpecÐPÑß 5Ñ, where 5 is the topology with basis f œ Öcc+ ± + − P× is called the Stone prime ideal space or the Stone dual space of P. Theorem 10.8 implies that if P is a Boolean algebra, then U œ f so the Stone and Priestley prime ideal topologies coincide. We will show in a moment that Duality 221 the Priestley prime ideal space of a bounded distributive lattice P is an ordered Boolean space. However, we leave it as an exercise to show that for nonBoolean distributive lattices, the Stone prime ideal space is not Hausdorff and so it is not a Boolean space. This is why the Priestley prime ideal topology is more suitable to the study of arbitrary bounded distributive lattices. Theorem 10.9 1) The Priestley prime ideal space ÐSpecÐPÑß 7 ß © Ñ of a bounded distributive lattice P is an ordered Boolean space. 2) The Stone prime ideal space ÐSpecÐFÑß 7 Ñ of a Boolean algebra F is a Boolean space. Proof. For part 1), to see that 7 is totally order-separated, suppose that U © y T. Then there is an + − U Ï T and so T − cc+ and U  cc+ . Since cc+ œ cc+ß! is a clopen down-set, 7 is totally order-separated. To prove compactness, the Alexander subbasis lemma (Theorem A1.4) implies that it is sufficient to show that any cover h of SpecÐPÑ by subbasis elements has a finite subcover. Suppose that h œ Öcc. ± . − H× ∪ Öc/ ± / − I× is a cover of SpecÐPÑ by subbasic elements. To insure that H and I are nonempty, we can assume that ! − H since cc! œ g and " − I since c" œ g. Thus, if T is a prime ideal, then .  T for some . − H or / − T for some / − I . Put another way, there is no prime ideal containing H and disjoint from I . Equivalently, there is no prime ideal containing ÐHÓ and disjoint from ÒIÑ. But then the prime separation theorem (Theorem 8.5) implies that ÐHÓ and IÑ cannot be disjoint and so there is an B − ÐHÓ ∩ ÒIÑ. Hence, there exist ." ß á ß .8 − H and /" ß á ß /7 − I for which /" • â • / 7 Ÿ B Ÿ . " ” â ” . 8 If T  c/3 for all 3, then T − cc/3 for all 3 and so T − ,cc/3 œ ccÐ/" •â•/7 Ñ © ccÐ." ”â”.8 Ñ œ .cc.4 and so T − cc.4 for some 4. This shows that the family Öc/" ß á ß c/7 ß cc." ß á ß cc.8 × is a finite subcover of h . For part 2), if F is a Boolean algebra, then the Stone prime ideal space SpecÐFÑ is the same as with the Priestley prime ideal space SpecÐFÑ and so it is an ordered Boolean space. However, since SpecÐFÑ is the family of all maximal ideals of F , it is an antichain and so the partial order on SpecÐFÑ is equality. Hence, 222 Lattices and Ordered Sets bclo ÐSpecÐFÑÑ œ k clo ÐSpecÐFÑÑ and SpecÐFÑ is a Boolean space. Now we can prove that the image of the rho map 3À P ä bÐSpecÐPÑÑ is the set bclo ÐSpecÐPÑÑ of clopen down-sets, that is, all clopen down-sets have the form cc+ for some + − P. Theorem 10.10 Let P be a bounded distributive lattice. Let U œ Öcc+ß, ± +ß , − P× be the basis for the Priestley prime ideal space SpecÐPÑ. 1) (Clopen sets) The clopen sets k clo ÐSpecÐPÑÑ are precisely the finite unions of basis elements cc+ß, . 2) (Clopen down-sets) The clopen down-sets are bclo ÐSpecÐPÑÑ œ f œ ÖTc+ ± + − P× and so the rho map 3À P Ä bclo ÐSpecÐPÑÑ is surjective and therefore a lattice isomorphism. Proof. For part 1), any finite union of the clopen basic sets is a clopen set. Conversely, if O is clopen, then it is a union of basic sets. However, O is also compact and so it is a finite union of basic sets. For part 2), let M − H. For each N  H, we have N © y M and so if we take BN − N Ï M , then M − ccBN and N − cBN . The family ÖcBN ± N  H× is an open cover of the compact set H- and so it has a finite subcover ÖcB" ß á ß cB8 ×, where we have simplified the subscripts a bit. Hence, H - © c B" ∪ â ∪ c B 8 and so M − ccB" ∩ â ∩ ccB8 © H Thus, H is the union of sets ccB" ∩ â ∩ ccB8 œ ccÐB" •â•B8 Ñ in f . Since H is compact, it is the union of finitely many of these sets and so H œ cc+" ∪ â ∪ cc+8 œ ccÐ+" ”â”+8 Ñ The Priestley Duality The Priestley duality, shown in Figure 10.3, Duality f L M ρL S ρM Oclo(Spec(L)) Oclo(Spec(M)) (f-1)-1 Oclo g X Y κX κY Spec(Oclo(X)) BDLat 223 (g-1)-1 Spec(Oclo(Y)) OrdBool Figure 10.3 is a duality between the category BDLat of bounded distributive lattices, together with lattice Ö!ß "×-homomorphisms and the category OrdBool of ordered Boolean spaces with continuous monotone maps. We begin our discussion with a preliminary result. For an ordered Boolean space \ , let ,ÐBÑ œ ^cB ³ ÖH − b clo Ð\Ñ ± B  H× Theorem 10.11 1) If 0 À P Ä Q is a lattice Ö!ß "×-homomorphism, then the restricted induced inverse map 0 " À SpecÐQ Ñ Ä SpecÐPÑ satisfies, for +ß , − P, Ð0 " Ñ" Ðcc+ Ñ œ cc0 Ð+Ñ (10.12) Ð0 " Ñ" Ðc+ Ñ œ c0 Ð+Ñ Ð0 " Ñ" Ðcc+ß, Ñ œ cc0 Ð+Ñß0 Ð,Ñ 2) If 1À \ Ä ] is a continuous monotone map, then the restricted induced inverse map 1" À bclo Ð] Ñ Ä b clo Ð\Ñ satisfies Ð1" Ñ" Ð^c+ Ñ œ ^c1Ð+Ñ " " Ð1 Ñ Ð^+ Ñ œ ^1Ð+Ñ " " Ð1 Ñ Ð^c+ ∩ ^, Ñ œ ^c1Ð+Ñ ∩ ^1Ð,Ñ (10.13) 224 Lattices and Ordered Sets Proof. For part 1), if + − P, then Ð0 " Ñ" Ðcc+ Ñ œ ÖT − SpecÐQ Ñ ± 0 " ÐT Ñ − cc+ × œ ÖT − SpecÐQ Ñ ± +  0 " ÐT Ñ× œ ÖT − SpecÐQ Ñ ± 0 Ð+Ñ Â T × œ cc0 Ð+Ñ and the rest follows from the fact that inverse maps are Boolean homomorphisms. Part 2) is entirely analogous. The Categories and Functors Let BDLat be the category of bounded distributive lattices, together with lattice Ö!ß "×-homomorphisms. Let OrdBool be the category of ordered Boolean spaces with continuous monotone maps. The relevant functors are described in the next theorem. Theorem 10.14 The following are contravariant functors. 1) fÀ BDLat Ê OrdBool defined by f ÐPÑ œ SpecÐPÑ and f Ð0 Ñ œ 0 " À SpecÐQ Ñ Ä SpecÐPÑ for any P − BDLat and any Ö!ß "×-homomorphism 0 À P Ä Q . 2) bclo À OrdBool Ê BDLat defined by bclo Ð\Ñ œ b clo Ð\Ñ and bclo Ð1Ñ œ 1" À b clo Ð] Ñ Ä b clo Ð\Ñ for any \ − OrdBool and any continuous monotone map 1À \ Ä ] . Proof. For part 1), it is clear that fÐ0 Ñ œ 0 " is monotone. Also, 0 " is continuous since Theorem 10.11 implies that the inverse image Ð0 " Ñ" Ðcc+ß, Ñ of a basic open set cc+ß, is open. Finally, if 0 À P Ä Q and 1À Q Ä R , then f Ð1 ‰ 0 Ñ œ Ð1 ‰ 0 Ñ" œ 0 " ‰ 1" œ f Ð0 Ñ ‰ f Ð1Ñ and f Ð+Ñ œ +. Hence, f is a contravariant functor. For part 2), bclo Ð0 Ñ œ 0 " is a lattice homomorphism and 0 " ÐgÑ œ g and 0 " Ð] Ñ œ \ Also, if 0 À \ Ä ] and 1À ] Ä ^ , then bclo Ð1 ‰ 0 Ñ œ Ð1 ‰ 0 Ñ" œ 0 " ‰ 1" œ bclo Ð0 Ñ ‰ b clo Ð1Ñ and bclo Ð+Ñ œ +. Hence, b clo is a contravariant functor. Duality 225 The Priestley and Stone Representations On the lattice side, Theorem 10.10 implies that the rho map 3À P ¸ bclo ÐSpecÐPÑÑ is a lattice isomorphism from a bounded distributive lattice P to the family of all clopen down-sets of a certain ordered Boolean space, namely, the Priestley prime ideal space SpecÐPÑ. On the ordered Boolean space side, for a given ordered Boolean space \ , let ^cB œ ÖH − b clo Ð\Ñ ± B  H× It is clear that ^cB − SpecÐbclo Ð\ÑÑ. We define the kappa function ,À \ Ä SpecÐbclo Ð\ÑÑ by ,ÐBÑ œ ^cB We would like to show that , is an order-homeomorphism, where SpecÐbclo Ð\ÑÑ has the Priestley prime ideal topology, that is, the sets ccHßI œ Öˆ − SpecÐb clo Ð\ÑÑ ± H  ˆß I − ˆ× for Hß I − bclo Ð\Ñ, form a basis for the topology. Theorem A1.6 implies that it is sufficient to show that , is a continuous order isomorphism. To see that , is an order embedding, note that the order in \ is determined by the clopen down-sets and so BŸC Í ^C © ^B Í ^cB © ^cC Í ,ÐBÑ © ,ÐCÑ To show that , is continuous, it is sufficient to show that the inverse image of any basic open set is open. But ," ÐccHßI Ñ œ ÖB − \ œ ÖB − \ œ ÖB − \ œ ÖB − \ œHÏI ± ,ÐBÑ − ccHßI × ± ^cB − ccHßI × ± H  ^cB and I − ^cB × ± B − H and B  I× and since H and I are clopen, so is H Ï I . Finally, to see that , is surjective, the image ,Ð\Ñ is compact and therefore closed. We show that any nonempty basic open set meets ,Ð\Ñ, which implies that ,Ð\Ñ œ SpecÐbclo Ð\ÑÑ. For a nonempty proper basic open set ccHßI œ Öˆ − SpecÐb clo Ð\ÑÑ ± H  ˆß I − ˆ× 226 Lattices and Ordered Sets we have H © y I and so there is an B − H Ï I . But B − H implies that H  ^cB and B  I implies that I − ^cB and so ^cB − ccHßI . The fact that the individual rho and kappa maps are isomorphisms gives representation theorems for bounded distributive lattices and ordered Boolean spaces. Theorem 10.15 1) (Priestley representation theorem for bounded distributive lattices) Let P be a bounded distributive lattice. The rho map 3À P Ä bclo ÐSpecÐPÑÑ defined by 3Ð+Ñ œ cc+ is a lattice isomorphism and so 3À P ¸ b clo ÐSpecÐPÑÑ Thus, every bounded distributive lattice is isomorphic to the power set sublattice of clopen down-sets of a Priestley prime ideal space, namely, SpecÐPÑ. 2) Let \ be an ordered Boolean space. The kappa map ,À \ Ä SpecÐb clo Ð\ÑÑ defined by ,ÐBÑ œ ^cB is an order-homeomorphism and so ,À \ ¸ SpecÐbclo Ð\ÑÑ Thus, every ordered Boolean space is order-homeomorphic to the Priestley prime ideal space of a bounded distributive lattice, namely, bclo Ð\Ñ. The Natural Isomorphisms We have seen that the families 3 œ Ö3P À P ¸ b clo ÐSpecÐPÑÑ ± P is a bounded distributive lattice× and , œ Ö,\ À \ ¸ SpecÐb clo Ð\ÑÑ ± \ is an ordered Boolean space× are families of lattice isomorphisms and order-homeomorphisms, respectively. In fact, 3 and , are natural isomorphisms, in symbols, 3À MBDLat ¸ bclo ‰ f † and ,À MOrdBool ¸ f ‰ b clo † and so the categories BDLat and OrdBool are dual. Indeed, the first equation in (10.12) can be written in the form Duality 227 Ð0 " Ñ" ‰ 3P œ 3Q ‰ 0 and the first equation in (10.13) can be written Ð1" Ñ" ‰ ,\ œ ,] ‰ 1 and these equations are precisely the defining conditions for natural transformations, as shown in Figure 10.3, which we repeat here in Figure 10.4. L f ρL Oclo(Spec(L)) M P ρM (f-1)-1 g Q κP Oclo(Spec(M)) Spec(Oclo(P)) κQ (g-1)-1 Spec(Oclo(Q)) Figure 10.4 Theorem 10.16 (The Priestley duality) 1) The families † bclo ‰ f 3 œ Ö3P ×À M Ä and † f ‰ b clo , œ Ö,\ ×À M Ä are natural isomorphisms and so BDLat and OrdBool are dual categories. In particular, Ð0 " Ñ" ‰ 3P œ 3Q ‰ 0 for any lattice Ö!ß "×-homomorphism 0 À P Ä Q and Ð1" Ñ" ‰ ,\ œ ,] ‰ 1 for any continuous monotone map 1À \ Ä ] . 2) The functors f and bclo are full and faithful, that is, the maps fÀ homBDLat ÐPß Q Ñ Ä homOrdBool ÐSpecÐPÑß SpecÐQ ÑÑ and bclo À homOrdBool Ð\ß ] Ñ Ä homBDLat Ðb clo Ð\Ñß b clo Ð] ÑÑ are bijections. We can say a bit more about the connection between a lattice Ö!ß "×homomorphism 0 À P Ä Q and its image f Ð0 Ñ œ 0 " under the spectrum functor. Theorem 10.17 Let 0 À P Ä Q be a lattice Ö!ß "×-homomorphism, where P and Q are bounded distributive lattices. 1) 0 is injective if and only if f Ð0 Ñ œ 0 " is surjective. 228 Lattices and Ordered Sets 2) 0 is surjective if and only if f Ð0 Ñ œ 0 " is an order embedding. Proof. For part 1), if 0 À P ä Q is injective, then 0 À P ¸ 0 ÐPÑ is an isomorphism and so 0 " À SpecÐ0 ÐPÑÑ Ä SpecÐPÑ is surjective. Hence, if T − SpecÐPÑ, then there is an V − SpecÐ0 ÐPÑÑ for which 0 " ÐVÑ œ T . We need only show that there is a U − SpecÐQ Ñ for which U ∩ 0 ÐPÑ œ V , since then 0 " ÐUÑ œ 0 " ÐU ∩ 0 ÐPÑÑ œ 0 " ÐVÑ œ T To this end, M œ Æ Q V is a proper ideal of Q and J œ Å Q Ð0 ÐPÑ Ï VÑ is a filter of Q that is disjoint from M . Hence, by the prime separation theorem, there is a prime ideal U − SpecÐQ Ñ for which V © U and U ∩ J œ g, that is, U ∩ 0 ÐPÑ œ V, as desired. For the converse to part 1), suppose that 0 " is surjective. If 0 Ð+Ñ œ 0 Ð,Ñ for ,Ÿ y +, then the prime separation theorem for points implies that there is a T − SpecÐPÑ such that + − T and ,  T . Since 0 " is surjective, there is a U − SpecÐQ Ñ such that 0 " ÐUÑ œ T . But 0 Ð,Ñ œ 0 Ð+Ñ − 0 ÐT Ñ © U implies that , − 0 " ÐUÑ, which is false. Hence, 0 is injective. For part 2), if 0 is surjective, then E©F Í 0 " ÐEÑ © 0 " ÐFÑ and so 0 " is an order embedding. For the converse, if 0 " is an order embedding, then for any prime ideals T and U of Q , T ©U Í Í Í 0 " ÐT Ñ © 0 " ÐUÑ 0 " ÐT ∩ 0 ÐPÑÑ © 0 " ÐUÑ T ∩ 0 ÐPÑ © U Thus, assuming that 0 is not surjective, we can obtain a contradiction by finding prime ideals T and U for which T © y U but T ∩ 0 ÐPÑ © U. Since 0 is not surjective, there is an + − Q Ï 0 ÐPÑ and we need only show that there are prime ideals T and U for which + − Tß +ÂU and T ∩ 0 ÐPÑ © U or equivalently, Æ+ © Tß Å+ ∩ U œ g and T ∩ 0 ÐPÑ © U Now, if T ∩ 0 ÐPÑ and Å + are disjoint, then there is such a prime ideal U, so it is sufficient to show that there is a prime ideal T for which Duality Æ+ © T 229 and ÐT ∩ 0 ÐPÑÑ ∩ Å + œ g But Æ + is disjoint from 0 ÐPÑ ∩ Å +, since +  0 ÐPÑ and so T also exists. The Case of Boolean Algebras The Priestley duality between bounded distributive lattices and ordered Boolean spaces can easily be restricted to a duality between the category BoolAlg of Boolean algebras with Boolean homomorphisms and the category Bool of Boolean algebras with continuous maps, as shown in Figure 10.5. L f ρL pclo(Spec(L)) M ρM (f-1)-1 pclo(Spec(M)) X S g κX Spec(pclo(X)) pclo BoolAl Y κY (g-1)-1 Spec(pclo(Y)) Bool Figure 10.5 First, we note that BoolAlg is a subcategory of BDLat. Also, we have seen that any Boolean space can be thought of as an ordered Boolean space under equality. Thus, since all maps are monotone in this case, it follows that Bool is a subcategory of OrdBool. As to the functors, the spectrum functor sends a Boolean algebra F to the Stone prime ideal space SpecÐFÑ, which is a Boolean space according to Theorem 10.9. Also, fÐ0 Ñ œ 0 " is a Boolean homomorphism. As to the down-set functor, in a Boolean space \ the order is equality and so all sets are down-sets, that is, bclo Ð\Ñ œ k clo Ð\Ñ and so the restriction of the functor bclo to Bool is the functor k clo , sending \ to k clo Ð\Ñ and sending a continuous map 1À \ Ä ] to the induced inverse map 1" À k clo Ð] Ñ Ä k clo Ð\Ñ. Thus, we have the two contravariant functors fÀ BoolAlg Ê Bool and k clo À Bool Ê BoolAlg Moreover, we have seen that the Ö!ß "×-homomorphisms 3P are Boolean isomorphisms when P is a Boolean algebra. Also, the kappa maps ,\ are homeomorphisms of Boolean spaces. Thus, we arrive at the duality in Figure 10.5. 230 Lattices and Ordered Sets Theorem 10.18 1) (Stone representation theorem for Boolean algebras) Let F be a Boolean algebra. The rho map 3À F Ä k clo ÐSpecÐFÑÑ defined by 3Ð+Ñ œ cc+ is a Boolean algebra isomorphism and so 3À F ¸ k clo ÐSpecÐFÑÑ Thus, every Boolean algebra is isomorphic to the Boolean algebra of clopen subsets of a Boolean space, namely, SpecÐFÑ. 2) Let \ be a Boolean space. The kappa map ,À \ Ä SpecÐk clo Ð\ÑÑ is a homeomorphism and so ,À \ ¸ SpecÐk clo Ð\ÑÑ Thus, every Boolean space is homeomorphic to the Stone prime ideal space of a Boolean algebra, namely, k clo Ð\Ñ. 3) (Stone duality for Boolean algebras) The families † k clo ‰ f 3 œ Ö3P ×À M Ä and † f ‰ k clo , œ Ö,\ ×À M Ä are natural isomorphisms and so BoolAl and Bool are dual categories. In particular, Ð0 " Ñ" ‰ 3P œ 3Q ‰ 0 for any Boolean homomorphism 0 À P Ä Q and Ð1" Ñ" ‰ ,\ œ ,] ‰ 1 for any continuous map 1À \ Ä ] . 4) The functors f and k clo are full and faithful, that is, the maps fÀ homBoolAl ÐPß Q Ñ Ä homBool ÐSpecÐPÑß SpecÐQ ÑÑ and k clo À homBool Ð\ß ] Ñ Ä homBoolAl Ðk clo Ð\Ñß k clo Ð] ÑÑ are bijections. Note that in a Boolean prime ideal space SpecÐFÑ, the clopen sets are precisely the sets cc+ . Also, the open sets are precisely the unions of clopen sets and the closed sets are precisely the intersections of clopen sets. Applications According to the Priestley duality, we may think of any bounded distributive lattice P as the family b clo Ð\Ñ of clopen down-sets of a Priestley prime ideal space \ œ SpecÐPÑ. What is the advantage of doing this? The space \ is, in Duality 231 some sense, simpler than the lattice bclo Ð\Ñ of clopen down-sets. Thus, many lattice-theoretic concepts may have simpler descriptions in topological terms than in lattice terms. For instance, the Priestley representation makes it easy to show that any bounded distributive lattice P is a sublattice of a Boolean algebra and to describe, up to isomorphism, a minimal such Boolean algebra. We first identify P with its faithful representation 3ÐPÑ œ Öcc+ × œ b clo ÐSpecÐPÑÑ Now, bclo ÐSpecÐPÑÑ is a sublattice of the Boolean algebra k clo ÐSpecÐPÑÑ. Moreover, if U is a Boolean subalgebra of k clo ÐSpecÐPÑÑ containing bclo ÐSpecÐPÑÑ, then for any basic clopen set cc+ß, of SpecÐPÑ, we have cc+ß, œ cc+ ∩ c, œ cc+ ∩ Ðcc, Ñ- − U and so cc+ß, − U for all +ß , − P. Then Theorem 10.10 implies that any element of k clo ÐSpecÐPÑÑ, being a finite union of the basic clopen sets cc+ß, , belongs to U and so U œ k clo ÐSpecÐPÑÑ Thus, k clo ÐSpecÐPÑÑ is a minimal Boolean algebra containing b clo ÐSpecÐPÑÑ as a sublattice. Theorem 10.19 If P is a bounded distributive lattice, then k clo ÐSpecÐPÑÑ is a minimal Boolean algebra containing the isomorphic copy b clo ÐSpecÐPÑÑ of P as a sublattice. The Priestley representation can be used to prove certain lattice-theoretic results, such as the result above, and to describe certain lattice-theoretic notions via the topological notions. Here are some additional examples. Recall that the pseudocomplement +‡ of + − P is defined to be the maximum semicomplement of +, that is, +‡ œ maxÖB − P ± + • B œ !× if the maximum exists. We will need the following facts, discussed in an earlier exercise. Two lattice Ö!ß "×-epimorphisms 0 À P Ä Q and 1À P Ä R with domain P are kernel equivalent, written 0 ´ 1, if the corresponding congruence kernels )0 and )1 are equal, that is, if 0 Ð+Ñ œ 0 Ð,Ñ Í 1Ð+Ñ œ 1Ð,Ñ 232 Lattices and Ordered Sets This is easily seen to be an equivalence relation on the class EpiÐPÑ of Ö!ß "×epimorphisms with domain P. Theorem 10.20 Let Ð\ß 7 ß Ÿ Ñ be an ordered Boolean space, with clopen down-sets bop Ð\Ñ. 1) (Ideals of bclo Ð\Ñ) The union map h À \ Ðbclo Ð\ÑÑ ¸ bop Ð\Ñ defined by h ÐˆÑ œ . ˆ is a lattice isomorphism and so bop Ð\Ñ is isomorphic to its own ideal lattice. Thus, any bounded distributive lattice is isomorphic to its ideal lattice. 2) (Pseudocomplements in bclo Ð\Ñ) The pseudocomplemented elements Y in bclo Ð\Ñ are precisely the elements Y for which Å Y is clopen, in which case Y ‡ œ \ Ï Å Y . 3) (Congruence relations on bclo Ð\Ñ) The lattice ConÐb clo Ð\ÑÑ of congruence relations on bclo Ð\Ñ is isomorphic to the lattice of open subsets of \ . Proof. We prove the third statement, leaving the others for the exercises. Let P œ bclo Ð\Ñ. Note that ÖB× is a closed set for all B − \ , since \ is Hausdorff. Also, Å B and Æ B are closed by Theorem 10.7. The plan is to find a bijection >Ð\Ñ Ç ConÐPÑ from the family >Ð\Ñ of closed sets of \ onto the set ConÐPÑ. Then we will show that this bijection is a lattice anti-isomorphism. We can then compose this map with the complement map to get a lattice isomorphism from the open sets onto ConÐPÑ. To find the bijection mentioned above, we first find a complete system of distinct representatives Ö0O ± O − >Ð\Ñ× for the kernel equivalence classes of EpiÐPÑ, indexed by the family >Ð\Ñ of closed sets. Then the map O Ä 0O Ä )0O where )1 is the congruence kernel of 1, is a bijection from >Ð\Ñ to ConÐPÑ, since every congruence relation ) is a congruence kernel, namely, the congruence kernel of the projection 1) . For each closed set O (with the subspace topology), consider the intersection by O map Duality 233 0O À bclo Ð\Ñ Ä b clo ÐOÑ defined by 0O ÐHÑ œ H ∩ O for any H − bclo Ð\Ñ. This map is a lattice Ö!ß "×-homomorphism and we leave it as an exercise to show that 0O is surjective, that is, 0O − EpiÐPÑ. We claim that the maps Ö0O ± O − bclo Ð\Ñ× form a complete system of distinct representatives for EpiÐPÑÎ ´ . If O" Á O# are distinct closed subsets of \ , then 0O" and 0O# are not kernel equivalent. To see this, suppose that O" © y O# clo and let B − O" Ï O# . We show that there are Y ß Z − b Ð\Ñ for which 0O# ÐY Ñ œ 0O# ÐZ Ñ and 0O" ÐY Ñ Á 0O" ÐZ Ñ Y ∩ O# œ Z ∩ O # and Y ∩ O" Á Z ∩ O" that is, If we find Y and Z for which B  Y and B − Z , then B  Y ∩ O" and B − Z ∩ O" and so Y ∩ O" Á Z ∩ O" . Thus, it is sufficient to show that B  Yß and Y ∩ O# œ Z ∩ O# B−Z or, equivalently, Å B ∩ Y œ gß ÆB © Z and Y ∩ O# œ Z ∩ O# (10.21) Now, about the only thing we can separate is O# ∩ Æ B and ÅB O# ∩ Å B and ÆB or since B  O# . Choosing the former, there is a clopen down-set Y for which O# ∩ Æ B © Y and Y ∩ ÅB œ g If we can find a clopen down-set Z that contains Y as well as Æ B, then Y ∩ O# © Z ∩ O# and we only arrange it so that Z ∩ O# © Y , or equivalently, that Z ∩ ÐO# Ï Y Ñ œ g. So we want to separate Y ∪ Æ B from O# Ï Y , which is possible since no element of O# Ï Y is less than or equal to any element of Æ B ∪ Y . Hence, there is a clopen down-set Z for which ÆB ∪ Y © Z and Z ∩ ÐO# Ï Y Ñ œ g and so Y ∩ O# œ Z ∩ O# . Thus, (10.21) holds. 234 Lattices and Ordered Sets It follows that for closed subsets O" and O# of \ , O" œ O # Í 0O" ´ 0O# and so each kernel equivalence class has at most one member of the form 0O . Now suppose that 0 À bclo Ð\Ñ Ä b clo Ð^Ñ is a Ö!ß "×-epimorphism, where ^ is an ordered Boolean space. Since the functor b clo is full, there is a continuous monotone map 2À ^ Ä \ for which 2" œ 0 . Since 0 is surjective, Theorem 10.17 implies that 2 is a continuous order embedding. Now, for Hß I − bclo Ð\Ñ, we have 0 ÐHÑ œ 0 ÐIÑ Í 2 " ÐHÑ œ 2 " ÐIÑ Í 2" ÐH ∩ 2Ð^ÑÑ œ 2 " ÐI ∩ 2Ð^ÑÑ But 2À ^ Ä 2Ð^Ñ is an order isomorphism and so the last equation above holds if and only if H ∩ 2Ð^Ñ œ I ∩ 2Ð^Ñ Finally, since 2 is continuous and ^ is compact, the image 2Ð^Ñ is compact and therefore closed in \ . Thus, we have shown that 0 ÐHÑ œ 0 ÐIÑ Í 02Ð^ Ñ ÐHÑ œ 02Ð^ Ñ ÐIÑ where 2Ð^Ñ is closed in \ . Thus, any Ö!ß "×-epimorphism 0 is equivalent to a Ö!ß "×-epimorphism of the form 0O for some closed set O in \ . This shows that the family Ö0O ± O − >Ð\Ñ× is a complete system of distinct representatives for kernel equivalence. Hence, the map CÀ >Ð\Ñ Ä ConÐPÑ defined by CÐOÑ œ )0O is a bijection. Moreover, if O" © O# then Y ∩ O# œ Z ∩ O # Ê Y ∩ O" œ Z ∩ O " 0O# ÐY Ñ œ 0O# ÐZ Ñ Ê 0O" ÐY Ñ œ 0O" ÐZ Ñ that is, But we have already shown that if O" © y O# , then there are clopen down-sets Y and Z for which Y ∩ O# œ Z ∩ O # and Y ∩ O" Á Z ∩ O" 0O# ÐY Ñ œ 0O# ÐZ Ñ Ê y and so Thus, 0O" ÐY Ñ œ 0O" ÐZ Ñ Duality Ð0O# ÐY Ñ œ 0O# ÐZ Ñ Ê 0O" ÐY Ñ œ 0O" ÐZ ÑÑ Í 235 O" © O# In terms of congruence relations, this is )0O# © )0O# Í O" © O # which shows that the bijection C is order-reversing and therefore a lattice antiisomorphism. Now, the complement map #À E È E- on \ is an latttice antiisomorphism from the lattice 7 of open sets in \ onto >Ð\Ñ and so the composition C ‰ # À 7 Ä ConÐPÑ is a lattice isomorphism. Exercises 1. 2. Let \ be totally separated. Prove that for any two disjoint compact sets E and F in \ , there is a clopen set G such that E © G and F ∩ G œ g. Let \ be totally order-separated. Prove that if E and F are disjoint compact sets with the property that no element of F is less than any element of E, that is, F ∩ Ð Æ EÑ œ g 3. 4. 5. 6. 7. then there is a clopen down-set G such that E © G and F ∩ G œ g. Let \ be an ordered Boolean space. Verify the following statements. a) The closed down-sets of \ are precisely the intersections of clopen down-sets. b) The open down-sets of \ are precisely the unions of clopen down-sets. Let Ð\ß 7 ß Ÿ Ñ be an ordered Boolean space. Prove that if G is closed in \ then Å G and Æ G are also closed. Let P be a bounded distributive lattice. Prove that + − P has a complement in P if and only if c+ is a down-set. a) Prove that the Cantor set V is an ordered Boolean space under the order inherited from ‘. b) Prove that bclo ÐVÑ is a countable Boolean algebra with no atoms. Prove that the intersection maps 0O À bclo Ð\Ñ Ä b clo ÐOÑ defined by 0O ÐHÑ œ H ∩ O 8. for any H − bclo Ð\Ñ are surjective. (Characterizing ideals in a bounded distributive lattice) Let Ð\ß 7 ß Ÿ Ñ be an ordered Boolean space. Let bop Ð\Ñ denote the family of open downsets of \ . Consider the maps h À \ Ðbclo Ð\ÑÑ Ä b op Ð\Ñ defined by 236 Lattices and Ordered Sets h ÐˆÑ œ . ˆ and i À b op Ð\Ñ Ä \ Ðb clo Ð\ÑÑ defined by i ÐHÑ œ ÖE − b clo Ð\Ñ ± E © H× In words, h ÐˆÑ is the union of all of the elements (clopen down-sets) of ˆ and iÐHÑ is the set of all clopen down-sets contained in H. a) Show that h ÐˆÑ is an open down-set in \ . b) Show that i ÐHÑ is an ideal of bclo Ð\Ñ. c) Show that and h ‰i œ+ 9. i ‰h œ+ where + is the appropriate identity function. d) Show that h is a lattice isomorphism. (Characterizing pseudocomplemented elements of a bounded distributive lattice) Let P œ bclo Ð\Ñ. Prove that the pseudocomplemented elements Y − bclo Ð\Ñ are precisely the elements Y for which Å Y is clopen, in which case Y ‡ œ \ Ï Å Y . Hint: Suppose that Y − bclo Ð\Ñ has a pseudocomplement Y ‡ , as shown in Figure 10.6. ↑U x U U* X Figure 10.6 What happens if there is an B − Ð Å Y Ñ- Ï Y ‡ ? Use Exercise 4. The Stone space of a bounded distributive lattice Definition Let P be a bounded distributive lattice. The set SpecÐPÑ together with the topology 5 with basis f œ Öcc+ ± + − P× is called the Stone prime ideal space or the Stone dual space of P. 10. Show that for a Boolean algebra, the Stone and Priestley prime ideal spaces are the same. 11. Prove that the Stone prime ideal space is compact. Duality 237 12. Prove that the open sets of the Stone prime ideal space are the sets ccM ³ ÖT − SpecÐPÑ ± M © y T× for M − \ ÐPÑ. 13. Prove that the compact open sets of the Stone prime ideal space are the basis elements Öcc+ ± + − P× 14. Prove that the Stone prime ideal space is Hausdorff if and only if P is a Boolean algebra. Consequently, the Stone prime ideal space of a nonBoolean bounded distributive lattice is not Hausdorff and so not a Boolean (Stone) space. Chapter 11 Free Lattices Lattice Identities Let us recall a few definitions from earlier in the book. Definition Let \ be a nonempty set. A lattice term (or just term), also called a lattice polynomial, over \ is any expression defined as follows. Let “ and ’ be formal symbols. 1) The elements of \ are terms of weight ". 2) If : and ; are terms, then so are Ð: ’ ;Ñ and Ð: “ ;Ñ whose weights are the sum of the weights of : and ; . 3) An expression in the symbols \ ∪ Ö ’ ß “ ß Ðß Ñ× is a term if it can be formed by a finite number of applications of 1) and 2). We denote the set of all lattice terms over \ by g\ . The elements of \ are called variables. Two terms are equal if and only if they are identical. Thus, for example, B" “ B# is not equal to B# “ B" . It is customary to omit the final pair of parentheses when writing lattice terms. A lattice term involving some or all of the variables B3" ß á ß B38 − \ but no others is denoted by :ÐB3" ß á ß B38 Ñ. We now define two binary operations • and ” , called meet and join respectively, on g\ . For :ß ; − g\ , let :•; œ:’; and : ” ; œ : “ ; Then since B3 ’ B4 œ B3 • B4 for all B3 ß B4 − \ and similarly for join, we can write any lattice term : − g\ using only the variables and the meet and join symbols (and parentheses). S. Roman (ed.), Lattices and Ordered Sets, doi: 10.1007/978-0-387-78901-9_11, © Steven Roman 2008 239 240 Lattices and Ordered Sets Note that while the set g\ of all terms over \ is not a lattice, it is an H-algebra, where H consists of the two binary operations of meet and join. Recall also that given a lattice P and a function 0 À \ Ä E , where E © P, the evaluation map %0 À g\ Ä P is defined, for each :ÐB3" ß á ß B38 Ñ − g\ by %0 :ÐB3" ß á ß B38 Ñ œ :Ð0 B3" ß á ß 0 B38 Ñ Definition A lattice identity or lattice equation over a nonempty set \ is an equation between lattice terms :ÐB" ß á ß B8 Ñ œ ;ÐB" ß á ß B8 Ñ where B3 − \ . Free and Relatively Free Lattices Now, given a nonempty set \ , we wish to describe the construction of the most general lattice P generated by the elements of \ (or a copy of these elements). This is called the free lattice generated by \ . By most general, we mean that a lattice identity should hold in J if and only if it holds in all lattices. Such a lattice can be characterized, up to isomorphism, by a universal mapping property. However, before giving this property, we want to generalize this notion a bit. A lattice is a set with two algebraic operations (meet and join) that satisfy certain lattice identities, namely, the associative, commutative, idempotent and absorption laws. If we add the modular law, we get modular lattices. If we add the distributive law, we get distributive lattices. More generally, we have the following concept. Definition Let D be a set of lattice equations. An equational class or variety based on D is the family ^ of all lattices that satisfy the equations in D. The lattices in ^ are called ^-lattices. The trivial variety is the variety for which D œ ÖB œ C× and so the ^-lattices are precisely the one-element lattices. The family of all lattices is an equational class with D œ g or D œ ÖB œ B×.. Also, the families of modular lattices and distributive lattices are equational classes. Definition Let ^ be an equational class. A lattice identity :ÐB" ß á ß B8 Ñ œ ;ÐB" ß á ß B8 Ñ holds in all ^-lattices if %0 :ÐB" ß á ß B8 Ñ œ %0 ;ÐB" ß á ß B8 Ñ Free Lattices 241 that is, :Ð0 B" ß á ß 0 B8 Ñ œ ;Ð0 B" ß á ß 0 B8 Ñ for all ^-lattices P and all functions 0 À \ Ä P. Now, given an equational class ^, we wish to describe the most general ^lattice J , in the sense that any lattice equation that holds in J also holds in all ^-lattices. Definition Let ^ be an equational class and let \ be a nonempty set, whose elements are called variables. Referring to Figure 11.1 κ X F τ f L Figure 11.1 we say that the pair ÐJ ß ,Ñ where J is a ^-lattice and ,À \ Ä J is a function is ^-universal if for any function 0 À \ Ä P, where P is a ^-lattice, there is a unique lattice homomorphism 70 À J Ä P for which 70 ‰ , œ 0 We will refer to the map 70 as the mediating morphism for 0 . If such a pair exists, then J is called a ^-free lattice and is said to be ^-freely generated by \ or have basis \ . The ^-free lattice J is denoted by FL^ Ð\Ñ. A lattice is relatively free if it is a ^-free lattice for some equational class ^. If D œ g, then a ^-free lattice FL^ Ð\Ñ is called a free lattice, is denoted by FLÐ\Ñ and is said to be freely generated by \ . If the equational class ^ is nontrivial, then the map ,À \ Ä FL^ Ð\Ñ in the definition above must be injective. To see this, let P be a ^-lattice with two distinct elements +ß , − P. If B Á C in \ , then we can define a function 0 À \ Ä P for which 0 ÐBÑ œ + and 0 ÐCÑ œ , . It follows that ,B Á ,C . Thus, if ^ is nontrivial, the image ,Ð\Ñ is a faithful copy of \ in FL^ Ð\Ñ. To see that the universal mapping property captures the spirit of ^-free lattices, we have the following. Theorem 11.1 Let \ be a nonempty set and let ÐJ ß ,À \ Ä J Ñ be a ^universal pair. Then a lattice identity 242 Lattices and Ordered Sets :ÐB" ß á ß B8 Ñ œ ;ÐB" ß á ß B8 Ñ over \ holds in all ^-lattices if and only if it holds in the ^-free lattice J , that is, if and only if :Ð,B" ß á ß ,B8 Ñ œ ;Ð,B" ß á ß ,B8 Ñ in J . Proof. One direction is clear so assume that the identity holds in the ^-free lattice J . Let P be a ^-lattice and let 0 À \ Ä P. If 7 À J Ä P is the mediating morphism for 0 , then 7 :Ð,B" ß á ß ,B8 Ñ œ 7 ;Ð,B" ß á ß ,B8 Ñ that is, :Ð7,B" ß á ß 7,B8 Ñ œ ;Ð7,B" ß á ß 7,B8 Ñ and so :Ð0 B" ß á ß 0 B8 Ñ œ ;Ð0 B" ß á ß 0 B8 Ñ that is, %0 :ÐB" ß á ß B8 Ñ œ %0 ;ÐB" ß á ß B8 Ñ Since this holds for all ^-lattices P and all evaluation maps %0 À \ Ä P, it follows that : œ ; holds in all ^-lattices. Before dealing with the existence of universal pairs, let us consider the issue of uniqueness. Theorem 11.2 Let ÐJ ß ,À \ Ä J Ñ and ÐPß 0 À \ Ä PÑ be ^-universal pairs. Then there is a lattice isomorphism .À J ¸ P for which .‰, œ0 In particular, J and P are isomorphic. For this reason, one often refers to the ^-free lattice on \ . Proof. Referring to Figure 11.2, the maps 7 À J Ä P and 5À P Ä J are mediating morphisms and 0 œ7 ‰, and , œ5‰0 0 œ Ð7 ‰ 5 Ñ ‰ 0 and , œ Ð5 ‰ 7 Ñ ‰ , Hence, However, the map 5 ‰ 7 À J Ä J and the identity map +À J Ä J are both mediating morphisms as well (as shown in the final diagram in the figure) and so the uniqueness requirement implies that 5 ‰ 7 œ +. Similarly, 7 ‰ 5 œ + and so . œ 7 is an isomorphism. Free Lattices κ X F f X τ L σ κ f κ X F στ=ι κ F L 243 F Figure 11.2 Constructing a Relatively Free Lattice If g\ is the family of lattice terms over \ and ,À \ Ä g\ is the inclusion map, then the pair Ðg\ ß ,Ñ almost fits the definition of a ^-free lattice. In particular, for any 0 À \ Ä P and any B − \ , we have %0 ‰ ,ÐBÑ œ 0 ÐBÑ where %0 is the evaluation map associated to 0 . Moreover, %0 preserves meets and joins. The problem is that g\ is not a ^-lattice! To fix this problem, we pass to equivalence classes. Define a binary relation on g\ by setting :) ; if : œ ; holds in all ^-lattices It is clear that this is an equivalence relation. Moreover, if :" );" and :# );# , then for any 0 À \ Ä P, where P is a ^-lattice, we have %0 :" œ %0 ;" and %0 :# œ %0 ;# and so %0 Ð:" ” :# Ñ œ %0 :" ” %0 :# œ %0 ;" ” %0 ;# œ %0 Ð;" ” ;# Ñ Hence, :" );" and :# );# Ê Ð:" ” :# Ñ)Ð;" ” ;# Ñ :" );" and :# );# Ê Ð:" • :# Ñ)Ð;" • ;# Ñ and similarly, These two properties can be expressed by saying that ) is a congruence relation on the H-algebra g\ . Theorem 11.3 Let \ be a nonempty set. 1) The set J œ g\ Î) is a ^-lattice under meet and join defined by Ò:Ó • Ò;Ó œ Ò: • ;Ó respectively. and Ò:Ó ” Ò;Ó œ Ò: ” ;Ó 244 Lattices and Ordered Sets 2) The pair Ðg\ Î ) ß , À X Ä g \ Î ) Ñ where ,ÐBÑ œ ÒBÓ, is universal and so g\ Î) is a ^-free lattice on \ . Proof. For part 1), to see that meet and join are well defined, if Ò:Ó œ Ò:w Ó and Ò;Ó œ Ò; w Ó, then :):w and ; ); w and so Ð: • ;Ñ)Ð:w • ; w Ñ, that is, Ò: • ;Ó œ Ò: w • ; w Ó and similarly for the join. Moreover, J is a ^-lattice since the axioms of a lattice and the additional lattice equations that define ^ hold in all ^-lattices. For example, since ÐÐ: • ;Ñ • <Ñ)Ð: • Ð; • <ÑÑ we have ÒÐ: • ;Ñ • <Ó œ Ò: • Ð; • <ÑÓ and so ÐÒ:Ó • Ò;ÓÑ • Ò<Ó œ ÒÐ: • ;Ñ • <Ó œ Ò: • Ð; • <ÑÓ œ Ò:Ó • ÐÒ;Ó • Ò<ÓÑ For part 2), if 0 À \ Ä P, where P is a ^-lattice, then the mediating morphism condition is 70 ÐÒBÓÑ œ 0 B for all B − \ . Now, the map 7 À J Ä P defined by 7 ÐÒ:ÐB" ß á ß B8 ÑÓÑ œ %0 Ð:ÐB" ß á ß B8 ÑÑ œ :Ð0 B" ß á ß 0 B8Ñ is well defined, since %0 is constant on the congruence classes of J . Also, 7 ÐÒ: • ;ÓÑ œ %0 Ð: • ;Ñ œ %0 Ð:Ñ • %0 Ð;Ñ œ 70 ÐÒ:ÓÑ • 7 ÐÒ;ÓÑ and similarly for join. Thus, since 7 ÐÒBÓÑ œ 0 B, it follows that 7 is a mediating morphism for 0 . As to uniqueness, if 5À J Ä P is also a lattice homomorphism for which 5 ‰ , œ 0 , that is, if 5ÐÒBÓÑ œ 0 B for all B − \ , then 7 ÐÒBÓÑ œ 0 B œ 5ÐÒBÓÑ and so 7 : œ 5: for any element :ÐÒB" Óß á ß ÒB8 ÓÑ of J , whence 7 œ 5. It is customary to drop the congruence class notation and denote the elements of FL^ Ð\Ñ œ g\ Î) as lattice terms and to refer to , as inclusion. Then we refer to the elements of g\ as formal lattice terms. Of course, it must be kept in mind that distinct formal lattice terms may be the same element of the ^-free lattice FL^ Ð\Ñ. Thus, B" ” B# and B# ” B" are distinct elements of g\ , but are the same elements of FL^ Ð\Ñ by the commutative law. Also, one must carefully check that functions defined on g\ are well defined when thought of as functions on FL^ Ð\Ñ. Free Lattices 245 Characterizing Equational Classes of Lattices In 1935, Garrett Birkhoff characterized equational classes of lattices among all families of lattices. Theorem 11.4 (Birkhoff, 1935 [7]) Let ^ be a family of lattices. The following are equivalent: 1) ^ is an equational class 2) ^ is closed under the taking of sublattices, homomorphic images and direct products 3) ^ is either trivial or else ^ is closed under homomorphic images and for every nonempty set \ , there is a ^-free lattice FL^ Ð\Ñ. Proof. It is not hard to see that 1) implies 2). Suppose that 2) holds and let ^ be nontrivial. Let FLÐ\Ñ œ g\ Î) be the free lattice on \ , where ) is the congruence relation on g\ defined by if : œ ; holds in all lattices :) ; Now consider the family Y œ Ö. − ConÐJ Ñ ± J Î. − ^× To see that Y is nonempty, note that since ^ is closed under sublattices and homomorphic images, it contains all one-element lattices. Hence, the relation . defined by +., for all +ß , − J has the property that J Î. − ^. The smallest member of Y is the meet - œ 4Ö. − ConÐJ Ñ ± J Î. − ^× and Theorem 9.17 implies that J Î- is isomorphic to a sublattice of the direct product $ÖJ Î. ± J Î. − ^× and so J Î- − ^. To see that J Î- is ^-free on \ , consider Figure 11.3, where 0 À \ Ä P and P − ^ and 4À \ Ä J is the inclusion map. j X f F f' π F/λ τ L Figure 11.3 246 Lattices and Ordered Sets The universality of the free lattice implies that there is a unique lattice homomorphism 0 w À J Ä P. Therefore, by the first isomorphism theorem, J ÎkerÐ0 w Ñ is isomorphic to a sublattice of P and so J ÎkerÐ0 w Ñ − ^, which implies that - © kerÐ0 w Ñ and so the universal mapping property of the quotient implies that there is a unique lattice homomorphism 7 À J Î- Ä P for which 7 ‰1‰4œ0 Thus, the pair ÐJ Î-ß 1 ‰ 4Ñ is ^-universal and so J Î- is ^-free on \ . Thus, 3) holds. Finally, suppose that 3) holds and continue the notation above with reference to Figure 11.3. Theorem 11.1 implies that a lattice identity : œ ; holds in a given ^-free lattice if and only if : œ ; holds in all ^-lattices. Now, let \! œ ÖB" ß B# ß á × be a countably infinite set of variables and let D be the set of all lattice identities in \! that hold in FL^ Ð\! Ñ. Let XD be the equational class associated to D. We want to show that ^ œ XD . First, the identities in D are over \! and hold in FL^ Ð\! Ñ and therefore also hold in all ^-lattices. Hence, ^ © XD . For the reverse inclusion, let P − XD , that is, all of the identities in D hold in P. Referring to Figure 11.4, let ] be a generating set for P. The function 0 À g] Ä P is defined by 0 :ÐC" ß á ß C8 Ñ œ :ÐC" ß á ß C8 Ñ where the :ÐC" ß á ß C8 Ñ on the left is a lattice term in g] , whereas the :ÐC" ß á ß C8 Ñ on the right is an element of P. TY f π TY/θ = FLK(Y) τ L Figure 11.4 If :ÐC" ß á ß C8 Ñ œ ;ÐC" ß á ß C8 Ñ in the ^-free lattice FL^ Ð] Ñ œ g] Î), then : œ ; in all ^-lattices and so :ÐB" ß á ß B8 Ñ œ ;ÐB" ß á ß B8 Ñ in FL^ Ð\! Ñ, that is, :ÐB" ß á ß B8 Ñ œ ;ÐB" ß á ß B8 Ñ is in D and so : œ ; holds in P. In the notation of Figure 11.4, if 1: œ 1; , then 0 : œ 0 ; , that is, ) œ kerÐ1Ñ © kerÐ0 Ñ and so the universality of the quotient implies that there is a unique lattice homomorphism 7 À FL^ Ð] Ñ Ä P for which 7 ‰ 1 œ 0 . Since 0 is surjective, so is 7 and so P is a homomorphic image of FL^ Ð] Ñ and is therefore a ^-lattice. Hence, P − ^ and so ^ œ XD . Thus, 1) holds. Free Lattices 247 The Word Problem for Free Lattices The word problem for ^-free lattices is the problem of determining, for any ^free lattice FL^ Ð\Ñ, when two formal lattice terms in \ represent the same element of FL^ Ð\Ñ. A solution to the word problem is an algorithm that makes this determination. If no such algorithm exists, then the word problem is said to be unsolvable. The word problem for free lattices (^ œ g) was first solved by the logician Thoralf Skolem in 1920 [56]. Skolem's solution was apparently not well known. In 1941, Whitman [65] produced a second solution to the word problem that is well known today. For other equational classes, the word problem is not necessarily solvable. In fact, Ralph Freese [18] proved in 1980 that there is no algorithm for deciding whether : œ ; in the free modular lattice generated by a set of size & and in 1982, Christian Herrmann [28] did the same for the free modular lattice generated by a set of size %. (Dedekind showed in 1900 [15] that the free modular lattice generated by a set of size $ has only #) elements.) For free lattices, Whitman's idea is to describe a decision procedure for all possible inequalities of the form : Ÿ ; among lattice terms in FLÐ\Ñ. Then, of course, we have : œ ; if and only if : Ÿ ; and ; Ÿ :. Theorem 11.5 Let FLÐ\Ñ be the free lattice on \ . Then the following hold, where Bß C − \ and :3 ß ;3 − FLÐ\Ñ for 3 œ "ß #: 1) (Variables are incomparable) BŸC Í BœC 2) (Variables are join prime) B Ÿ ;" ” ;# Í B Ÿ ;" or B Ÿ ;# Í :" Ÿ B or :# Ÿ B 3) (Variables are meet prime) :" • : # Ÿ B 4) (Definition of meet) : Ÿ ;" • ; # Í : Ÿ ;" and B Ÿ ;# Í :" Ÿ ; and :# Ÿ ; 5) (Definition of join) :" ” : # Ÿ ; 6) (Whitman's condition) : ³ :" • : # Ÿ ; ³ ; " ” ; # Í :" Ÿ ; or :# Ÿ ; or : Ÿ ;" or : Ÿ ;# 248 Lattices and Ordered Sets Proof. The “if” statements hold in all lattices. Also, statements 4) and 5) hold in all lattices. Next, we show that for any B − \ and C" ß á ß C7 − \ Ï ÖB×, BŸ y 2 C3 which, in particular, proves 1). To this end, define a function 0 À \ Ä #, where # œ Ö!ß "× is the lattice with two elements ! " by setting 0 B œ " and 0 C œ ! for all C − \ Ï ÖB×. This can be extended uniquely to a lattice homomorphism 70 À FLÐ\Ñ Ä #. But then B Ÿ 2C3 " œ 0 B Ÿ 20 C3 œ ! Ê which is false. Hence, B Ÿ y 1C3 . Next, we observe that all A − FLÐ\Ñ satisfy A Ÿ 2C3 for some C3 − \ Ï ÖB× 8 BŸA or 3œ" For if W is the set of elements of FLÐ\Ñ for which this holds, then \ © W and W inherits meets and joins from FLÐ\Ñ, whence W œ FLÐ\Ñ. Now we can prove 2). Suppose that B Ÿ ;" ” ;# . If B Ÿ y ;" and B Ÿ y ;# , then ; " Ÿ 2 C3 8 ; " Ÿ 2 D3 7 and 3œ" 3œ" for some C3 ß D3 − \ Ï ÖB× and so 2 C3 8 B Ÿ ;" ” ;# Ÿ 3œ" 2 D3 7 ” 3œ" which we have shown to be false. Statement 3) is the dual of 2). As to Whitman's condition, Alan Day [14] has provided a simple proof. Suppose that Whitman's condition fails in J œ FLÐ\Ñ for some elements : ³ :" • : # Ÿ ; ³ ; " ” ; # We construct a new lattice in which : Ÿ y ; , by replacing the interval M œ Ò:ß ;Ó in J with a pair of intervals that are distinct copies of M : M! œ M ‚ Ö!× œ ÖÐ?ß !Ñ ± ? − M× and M" œ M ‚ Ö"× œ ÖÐ?ß "Ñ ± ? − M× Let Free Lattices 249 J ‡ œ ÐJ Ï MÑ ∪ M" ∪ M# Extend the order on J Ï ÖM× to J ‡ by setting (for 3 œ !ß ") @ Ÿ Ð?ß 3Ñ in J ‡ Ð?ß 3Ñ Ÿ @ in J ‡ Ð?ß 3Ñ Ÿ Ð@ß 4Ñ in J ‡ Í Í Í @ Ÿ ? in J ? Ÿ @ in J ? Ÿ @ in J and 3 Ÿ 4 We leave it to the reader to check that J ‡ is a lattice under this order. Also, in J ‡ , we have :" • :# œ Ð:ß "Ñ Ÿ y Ð;ß !Ñ œ ;" ” ;# We can associate two maps with this construction. The first map 1À J ‡ Ä J is projection of J ‡ onto J defined by 1? œ ? for ? − J Ï M and 1Ð?ß 3Ñ œ ? for 3 œ !ß " This is a lattice epimorphism. For the second map, the injection 1À \ Ä J ‡ defined by 1ÐBÑ œ œ B ÐBß !Ñ for B  M for B − M can be extended to a lattice homomorphism 71 À J Ä J ‡ . Note that the composition 1 ‰ 71 À J Ä J satisfies 1Ð71 ÐBÑÑ œ B for all B − \ and so 1 ‰ 71 œ ", the identity on J . Now, the failure of Whitman's condition means that :" Ÿ y ; and :# Ÿ y ; and :Ÿ y ;" and : Ÿ y ;# and so :3 ß ;3  M œ Ò:ß ;Ó. Therefore, if = − Ö:" ß :# ß ;" ß ;# ×, then 1Ò71 Ð=ÑÓ œ = œ 1= and since 1 is injective on J Ï M , it follows that 71 Ð=Ñ œ =. Therefore, 71 Ð:Ñ œ : and 71 Ð;Ñ œ ; and so in J ‡ , we have 71 Ð:Ñ œ : Ÿ y ; œ 71 Ð;Ñ But : Ÿ ; in J implies that 71 Ð:Ñ Ÿ 71 Ð;Ñ in J ‡ , which is a contradiction. Thus, Whitman's condition holds. Theorem 11.5 provides a decision procedure for determining whether : œ ; is true in the free lattice FLÐ\Ñ. We simply apply the rules in the theorem to the inequalities : Ÿ ; and ; Ÿ :. Each application reduces the weight of the inequality and so must eventually terminate with a yes or no answer. For example, to decide whether B ” ÐC ” ÐB • DÑÑ Ÿ B ” C 250 Lattices and Ordered Sets we apply 5) to get BŸB”C and C ” ÐB • DÑ Ÿ B ” C The first inequality is true by 2). As to the second, another application of 5) gives C ŸB”C and B • D Ÿ B ” C the first of which is true. As to the second, Whitman's condition says that this holds since B • D Ÿ B by 4). Thus, the decision is yes. Canonical Forms The following concept is important in the study of free lattices. Definition Let E and F be finite subsets of a lattice P. If each + − E is contained in some , − F , then E refines F , denoted by E ¥ F . The dual notion is that every element of E contains an element of F , written E ¦ F . The basic properties of refinement are given in the exercises. We will require the following property, whose proof is an exercise. Lemma 11.6 If E ¥ F and F ¥ E, then E and F have the same maximal elements. Let \ be a nonempty set. Using the equivalence class notation, every element Ò>Ó of the free lattice FLÐ\Ñ œ g\ Î) is an equivalence class of terms in g\ and we refer to each element of Ò>Ó as a representative of Ò>Ó. We wish to show that the representatives of Ò>Ó of minimum weight are unique up to applications of the associativity and commutativity laws. For example, the following are the same up to associativity and commutativity, B • ÐC • DÑ and ÐC • BÑ • D Let us make this idea more precise. Every term > − g\ that is not a variable has an outermost operation (meet or join). If the outermost operation for > is the join, then > has the form > œ Ð?" Ñ ” Ð?# Ñ If the outermost operation of ?" is join, then we can write > œ ÐÐ?"ß" Ñ ” Ð?"ß# ÑÑ ” Ð?# Ñ and use the formal associativity law to remove parentheses: Ð?"ß" Ñ ” Ð?"ß# Ñ ” Ð?# Ñ If we continue to remove parentheses in this manner, the result is a join Free Lattices 251 Ð@" Ñ ” â ” Ð@8 Ñ where each @3 is either a variable or the outermost operation of @3 is meet. We can repeat the process with the terms @3 , resulting in ÐÐ@"ß" Ñ • â • Ð@"ß5" ÑÑ ” â ” ÐÐ@8ß" Ñ • â • Ð@8ß58 ÑÑ where each @3ß4 is either a variable or the outermost operation of @3ß4 is join. This process can be repeated until it cannot be continued further, resulting in a string of one of the following forms, which we refer to as normal form (by analogy with the disjunctive/conjunctive normal forms of Boolean algebra). Definition Let E œ \ ∪ Ö • ß ” ß Ðß Ñ× and let jE be the set of all strings over the alphabet E. An element of \ is called a variable. The weight AÐ=Ñ of a string = − jE is the number of occurrences of variables in =. 1) A member of \ has join-normal form and meet-normal form in jE . 2) Assume that join-normal form and meet-normal form have been defined for members of jE of weight less than 8 #. a) A string over E of weight 8 # has join-normal form if it has the form = œ Ð=" Ñ ” â ” Ð=5 Ñ where 5 # and each =3 has meet-normal form. b) A string over E of weight 8 # has meet-normal form if it has the form = œ Ð=" Ñ • â • Ð=5 Ñ where 5 # and each =3 has join-normal form. An element of j\ has normal form if it has either meet-normal or join-normal form. Variables have trivial normal form; all other normal-form strings have nontrivial normal form. Each term > − g\ can be reduced to a unique string NFÐ>Ñ in normal form, called the normal form of >. Note that NFÐ>Ñ is obtained from > by removing only parentheses. Hence, the occurrences and relative locations of the variables are the same in both expressions. It is also clear that any string in normal form can be “reverse engineered,” generally in more than one way, to produce a term in g\ . Also, the weights of > and NFÐ>Ñ are the same, that is, AÐ>Ñ œ AÐNFÐ>ÑÑ and their values as members of the free lattice FLÐ\Ñ are the same, that is, Ò>Ó œ ÒNFÐ>ÑÓ Having the same normal form expresses the fact that two terms are equivalent up to associativity. To deal with commutativity, we define an equivalence 252 Lattices and Ordered Sets relation ´ - on the family of normal-form strings by saying that two normalform strings = and > are equivalent if one can be obtained from the other by reordering the terms within zero or more join substrings or meet substrings Ð?" Ñ ” â ” Ð?5 Ñ or Ð?" Ñ • â • Ð?5 Ñ anywhere within = and >. Then we define an equivalence relation ´ on g\ by = ´ +- > if NFÐ=Ñ ´ - NFÐ>Ñ When = ´ +- >, we say that = and > are equivalent up to associativity and commutativity. We wish to show that if ? − Ò>Ó has minimum weight, then the minimum weight terms in Ò>Ó are the terms that are ´ +- -equivalent to ?. Since taking the normal form preserves weight, we need only show that any two terms of minimum weight are ´ +- -equivalent. For this, it is sufficient to show that if αß " − j\ are normal forms of minimum weight for elements of Ò>Ó, then α ´ - " . If ? − Ò>Ó has minimum weight, we can get some information about NFÐ?Ñ from the nature of Ò>Ó. If Ò>Ó œ ÖB× for some variable B − \ , then ? œ > œ B and NFÐ?Ñ œ B. If Ò>Ó œ 2Ò>3 Ó is a proper join in FLÐ\Ñ, that is, if Ò>3 Ó Ò>Ó for all 3, then NFÐ?Ñ must be a nontrivial join-normal form. For if NFÐ?Ñ œ B is a variable, then ÒBÓ œ Ò>Ó œ 2Ò>3 Ó and so ÒBÓ Ÿ Ò>5 Ó for some 5 , whence Ò>5 Ó œ 1Ò>3 Ó œ Ò>Ó, contradicting the fact that 1Ò>3 Ó is a proper join. Also, if NFÐ?Ñ has nontrivial meet-normal form NFÐ?Ñ œ 4Ð?4 Ñ then 4Ò?4 Ó œ ÒNFÐ?ÑÓ œ Ò?Ó œ Ò>Ó œ 2Ò>3Ó and Whitman's condition implies that one of the following must hold: 1) Ò?5 Ó Ÿ 1Ò>3 Ó for some 5 . Then Ò?5 Ó œ 3Ò?4 Ó œ Ò>Ó, which implies that ?5 − Ò>Ó. But AÐ?5 Ñ AÐNFÐ?ÑÑ œ AÐ?Ñ, which contradicts the fact that ? has minimum weight. 2) 3Ò?4 Ó Ÿ Ò>5 Ó for some 5 . But then Ò>5 Ó œ 1Ò>3 Ó œ Ò>Ó, contradicting the fact that 1Ò>3 Ó is a proper join. Free Lattices 253 Thus, NFÐ?Ñ has nontrivial join-normal form NFÐ?Ñ œ 2ÖÐ?3 Ñ ± 3 − M× If ? has minimum weight in Ò>Ó, there are several things we can say about the elements Ò?3 Ó of the free lattice FLÐ\Ñ. Theorem 11.7 Let > − g\ and let ? − Ò>Ó have minimum weight. 1) If Ò>Ó œ Ò>" Ó ” â ” Ò>8 Ó is a proper join in FLÐ\Ñ, then ? has a nontrivial join-normal form NFÐ?Ñ œ Ð?" Ñ ” â ” Ð?7 Ñ and the following properties hold in FLÐ\Ñ: P1) The Ò?3 Ó are distinct and Y œ ÖÒ?" Óß á ß Ò?7 Ó× is an antichain P2) If ?5 œ +" • â • +7 , then Ò+3 Ó Ÿ y Ò>Ó for all 3 P3) Each ?3 is a minimum weight representation of Ò?3 Ó. P4) Ö?" ß á ß ?7 × ¥ Ö>" ß á ß >8 × 2) Dually, if Ò>Ó œ Ò>" Ó • â • Ò>8 Ó is a proper meet in FLÐ\Ñ, then ? has a nontrivial meet-normal form NFÐ?Ñ œ Ð?" Ñ • â • Ð?7 Ñ and the following properties hold in FLÐ\Ñ: P1') œ P1) P2') If ?5 œ +" ” â ” +7 , then Ò>Ó Ÿ y Ò+3 Ó for all 3 P3') œ P3) P4') Ö?" ß á ß ?7 × ¦ Ö>" ß á ß >8 × Proof. If P1) fails, then there is a proper subset M! of M for which Ò?Ó œ 2ÖÒ?3 Ó ± 3 − M× œ 2ÖÒ?3 Ó ± 3 − M! × and then the string 2Ö?3 ± 3 − M! × can be reverse engineered to a term : − Ò?Ó œ Ò>Ó of smaller weight than that of ? and so P1) holds. For P2), if ?5 œ +" • â • +7 , then Ò+" Ó • â • Ò+7 Ó Ÿ Ò>" Ó ” â ” Ò>8 Ó and so Whitman's condition implies that Ò+4 Ó Ÿ Ò>Ó for some 4 or Ò?5 Ó Ÿ >4 for 254 Lattices and Ordered Sets some 4. But if Ò+4 Ó Ÿ Ò>Ó for some 4, then, Ò>Ó Ÿ Ò+4 Ó ” Š23Á5 Ò>3 Ó‹ Ÿ Ò>Ó and so Ò>Ó œ Ò+4 Ó ” Š23Á5 Ò>3 Ó‹ which implies that Ò>Ó has a shorter representation, obtained by reverse engineering the string + ” Š23Á5 >3 ‹ Thus, Ò+3 Ó Ÿ y Ò>Ó for all 3 which is P2). Note that P4) follows from P2) and Whitman's condition. P3) is clear. We can now show that minimum weight representatives are unique up to associativity and commutativity. Theorem 11.8 Let > − g\ . 1) If ?ß @ − Ò>Ó have minimum weight in Ò>Ó, then NFÐ?Ñ ´ - NFÐ@Ñ that is, ? and @ are equivalent up to associativity and commutativity. 2) a) If Ò>Ó is a proper join in FLÐ\Ñ, then @ − Ò>Ó has minimum weight if and only if its join-normal form NFÐ@Ñ œ Ð@" Ñ ” â ” Ð@7 Ñ satisfies properties P1)–P3), and therefore P4) of Theorem 11.7. b) If Ò>Ó is proper meet in FLÐ\Ñ, then @ − Ò>Ó has minimum weight if and only if its meet-normal form NFÐ@Ñ œ Ð@" Ñ • â • Ð@7 Ñ satisfies properties P1')–P3'), and therefore P4') of Theorem 11.7. Proof. We have seen that minimum weight representations satisfy properties P1)–P4) or P1')–P4'). For the converses of 2a) and 2b), let NFÐ@Ñ œ Ð@" Ñ ” â ” Ð@7 Ñ satisfy properties P1)–P3), and therefore P4), and let ? − Ò>Ó have minimum weight, with Free Lattices 255 NFÐ?Ñ œ Ð?" Ñ ” â ” Ð?8 Ñ Since Y œ ÖÒ?" Óß á ß Ò?8 Ó× and Z œ ÖÒ@" Óß á ß Ò@7 Ó× both satisfy P4), we have Y ¥Z and Z ¥Y and since Y and Z are antichains, Lemma 11.6 implies that Y œ Z . Thus, by reindexing if necessary, we have 7 œ 8 and Ò@3 Ó œ Ò?3 Ó for all 3. It follows that both @3 and ?3 are minimum weight representatives of Ò?3 Ó and so AÐ@3 Ñ œ AÐ?3 Ñ for all 3. Hence, AÐ@Ñ œ AÐ?Ñ is minimum. Definition If > − g\ , then the terms in Ò>Ó of mimimum weight are called canonical forms for Ò>Ó. Thus, the canonical forms for a term > − g\ are ´ +- -equivalent, that is, equivalent up to associativity and commutativity. The Free Lattice on Three Generators Is Infinite We wish to show that the free lattice FLÐ\Ñ on three generators \ œ ÖBß Cß D× has an infinite strictly increasing sequence, which shows that FLÐ\Ñ is infinite. We begin with an application of Theorem 11.5 to get some cancellation rules. Lemma 11.9 Let J\ be a free lattice over \ and let ?ß @ß A be distinct members of \ . Then the following cancellation rules hold for :ß ; − J\ : 1) ? Ÿ @ ” : Ê ? Ÿ : 2) @ • : Ÿ ? Ê : Ÿ ? 3) ? • Ð@ ” :Ñ Ÿ @ ” ÐA • ;Ñ Ê : Ÿ @ ” ÐA • ;Ñ 4) ? • Ð@ ” :Ñ Ÿ A ” Ð? • ;Ñ Ê ? • Ð@ ” :Ñ Ÿ ; We write these cancellation rules in the following abbreviated form, which shows the cancellations in bold: 1) ? Ÿ @ ” : 2) @ • : Ÿ ? 3) ? • Ð@ ” :Ñ Ÿ @ ” ÐA • ;Ñ 4) ? • Ð@ ” :Ñ Ÿ A ” Ð? • ;Ñ Proof. Parts 1) and 2) follow directly from Theorem 11.5. For part 3), Whitman's condition implies that one of the following must hold: 5) 6) 7) 8) ? Ÿ @ ” ÐA • ;Ñ @ ” : Ÿ @ ” ÐA • ;Ñ ? • Ð@ ” :Ñ Ÿ @ ? • Ð@ ” :Ñ Ÿ A • ; Now, 5) and 1) imply that ? Ÿ A • ; Ÿ A, which is false. Also, 8) implies that ? • Ð@ ” :Ñ Ÿ A and so 2) gives 256 Lattices and Ordered Sets @Ÿ@”: ŸA which is false. On the other hand, 6) implies that : Ÿ @ ” ÐA • ;Ñ which gives 3). Finally, 7) and 2) imply that @ ” : Ÿ @, that is, : Ÿ @ Ÿ @ ” ÐA • ;Ñ which is also 3). Thus, 3) holds. For part 4), Whitman's condition implies that one of the following must hold: 9) 10) 11) 12) ? Ÿ A ” Ð? • ;Ñ @ ” : Ÿ A ” Ð? • ;Ñ ? • Ð@ ” :Ñ Ÿ A ? • Ð@ ” :Ñ Ÿ ? • ; Now, 9) and 1) imply that ? Ÿ ? • ; Ÿ ; and so 4) holds in this case. Statement 10) implies that @ Ÿ A ” Ð? • ;Ñ and so 1) gives @ Ÿ ? • ; Ÿ ?, which is false. Statement 11) and 2) give @Ÿ@”: ŸA which is false. Finally, 12) implies that ? • Ð@ ” :Ñ Ÿ ; which is 4). Thus 4) holds in all cases. Now we can prove that FLÐÖBß Cß D×Ñ is infinite. Theorem 11.10 The free lattice FLÐ\Ñ on the set \ œ ÖBß Cß D× has an infinite strictly ascending chain and is therefore an infinite lattice. Proof. We use the cancellation rules 1) 2) 3) 4) 5) 6) ?Ÿ@”: @•: Ÿ? ? • Ð@ ” :Ñ Ÿ @ ” ÐA • ;Ñ ? • Ð@ ” :Ñ Ÿ A ” Ð? • ;Ñ ?Ÿ@•: @”: Ÿ? Free Lattices 257 the last two being obvious. Let :! œ B and for 8 !, let :8 œ B ” ÐC • ÐD ” ÐB • ÐC ” ÐD • :8" ÑÑÑÑÑ This is the alternating meet and join of Dß C and B in that order done twice. We first explore the relationship between the variables B, C and D and the elements :8 . Clearly, B Ÿ :" . Actually, B :" , for if :" Ÿ B, that is, if B ” ÐC • ÐD ” ÐB • ÐC ” ÐD • BÑÑÑÑÑ Ÿ B then cancellation rules 6) and 2) imply that D ” ÐB • ÐC ” ÐD • BÑÑÑ Ÿ B and so D Ÿ B, which is false. Hence, B :" . As a result, since the meet and join operations are monotone, the sequence Ð:8 Ñ is nondecreasing and so B :" Ÿ : 8 for all 8 ". Our eventual goal is to show that Ð:8 Ñ is strictly increasing. If C Ÿ :8 , that is, if C Ÿ C • ÐD ” ÐB • ÐC ” ÐD • :8" ÑÑÑÑ then cancellation rules 5) and 1) imply that C Ÿ B • ÐC ” ÐD • :8" ÑÑ Ÿ B which is false and so C Ÿ y :8 . But :8 Ÿ C implies that B Ÿ C , which is also false and so C ² :8 for all 8 !. If D Ÿ :8 , that is, if D Ÿ C • ÐD ” ÐB • ÐC ” ÐD • :8" ÑÑÑÑ then D Ÿ C, which is false. Also, :8 Ÿ D implies that B Ÿ D , which is false. Thus, B :" Ÿ :8 for 8 " and Cß D ² :8 for 8 ! Now suppose that 8 is the smallest integer for which :8" Ÿ :8 , that is, B ” ÐC • ÐD ” ÐB • ÐC ” ÐD • :8 ÑÑÑÑÑ Ÿ :8 then 6) gives C • ÐD ” ÐB • ÐC ” ÐD • :8 ÑÑÑÑ Ÿ :8 that is, 258 Lattices and Ordered Sets C • ÐD ” ÐB • ÐC ” ÐD • :8 ÑÑÑÑ Ÿ B ” ÐC • ÐDðóóóóóóóóóóóñóóóóóóóóóóóò ” ÐB • ÐC ” ÐD • : 8"ÑÑÑÑÑ ðóóóóóóñóóóóóóò : ; Making the substitutions indicated by the underbraces above gives C • ÐD ” :Ñ Ÿ B ” ÐC • ;Ñ and so cancellation rule 4) gives C • ÐD ” :Ñ Ÿ ; that is, C • ÐD ” ÐB • ÐC ” ÐD • :8 ÑÑÑÑ Ÿ D ” ÐB • Ððóóóóñóóóóò C ” ÐD • :8"Ñ ÑÑ ðóóóóóóñóóóóóóò : ; Again making the substitutions indicated by the underbraces above gives C • ÐD ” :Ñ Ÿ D ” ÐB • ;Ñ and so cancellation rule 3) gives : Ÿ D ” ÐB • ;Ñ that is, B • ÐC ” Ðï D • :8 ÑÑ Ÿ D ” ÐB • Ððóóóóñóóóóò C ” ÐD • :8" Ñ ÑÑ : ; Again making the substitutions indicated by the underbraces above gives B • ÐC ” :Ñ Ÿ D ” ÐB • ;Ñ and so cancellation rule 4) gives B • ÐC ” :Ñ Ÿ ; that is, B • ÐC ” Ðï D • :8 ÑÑ Ÿ C ” ÐD • î :8" Ñ : ; Making the substitutions indicated by the underbraces above gives B • ÐC ” :Ñ Ÿ C ” ÐD • ;Ñ and so cancellation rule 3) gives : Ÿ C ” ÐD • ;Ñ that is, D • :8 Ÿ C ” ÐD • :8" Ñ Free Lattices 259 An application of Whitman's condition implies that one of the following must hold: 7) 8) 9) 10) D Ÿ C ” ÐD • :8" Ñ :8 Ÿ C ” ÐD • :8" Ñ D • :8 Ÿ C D • :8 Ÿ D • :8" But 7) and 1) imply that D Ÿ D • :8" Ÿ :8" which is false. Also, 8) implies that B Ÿ :8 Ÿ C ” ÐD • :8" Ñ Ÿ C ” D which is false. Statement 9) implies that :8 Ÿ C , which is false. Finally, 10) implies that D • :8 Ÿ :8" By Whitman's condition, since :8" is a meet, we have four possibilities to consider: 1) 2) 3) 4) D Ÿ :8" :8 Ÿ :8" D • :8 Ÿ B D • :8 Ÿ C • ÐD ” ÐB • ÐC ” ÐD • :8# ÑÑÑÑ Now, 1) is false and 2) is false by the minimality of 8. Also, 3) implies that :8 Ÿ B, which is false. Finally, 4) implies that D • :8 Ÿ C which is also false. This contradiction implies that there is no integer 8 for which :8" Ÿ :8 . Free lattices can be quite complicated, as witnessed by the fact that a free lattice with $ generators is infinite. Moreover, it can be shown that a free lattice on $ generators contains a sublattice that is free on a countably infinite number of generators! We note also that the free lattice FLÐ\Ñ is not complete for k\ k $. The reader interested in more information about free lattices should consult the 1995 book Free Lattices, by Freese, Jezek and Nation [19]. Exercises 1. 2. If E ¥ F then does it follow that F ¦ E? Let E and F be finite subsets of a lattice P. Prove the following: a) E ¥ F implies 1E Ÿ 1F . 260 Lattices and Ordered Sets b) c) d) e) f) 3. 4. ¥ is a quasiorder on P. E © F implies E ¥ F . If E is an antichain and E ¥ F and F ¥ E, then E © F . If E and F are antichains and E ¥ F and F ¥ E, then E œ F . If E ¥ F and F ¥ E, then E and F have the same maximal elements. Describe the free lattice on the set \ œ ÖBß C×. (The free lattice on three elements is infinite.) Prove that the inequality + • Ð, ” -Ñ Ÿ Ð+ • ,Ñ ” Ð+ • -Ñ 5. 6. 7. does not hold in a free lattice. What conclusions do you draw about distributivity? Let FLÐ\Ñ be the free lattice on \ . a) Prove that every element of FLÐ\Ñ is either meet-irreducible or joinirreducible. b) Prove that the elements of FLÐ\Ñ that are both meet-irreducible and join-irreducible are the elements of \ . c) Prove that FLÐ\Ñ œ FLÐ] Ñ implies \ œ ] . Suppose that P is a lattice generated by a set \ . Suppose also that \ satisfies statements 1)–6) in Theorem 11.5. Show that P is free on \ . Hint: Apply the universal mapping property to the inclusion map 0 À \ Ä P. Show that the mediating morphism is an isomorphism. Let P be a lattice with generating set \ and assume that P satisfies Whitman's condition. Show that P is isomorphic to the free lattice FLÐ\Ñ if and only if the following hold for all B − \ and finite subsets ] © \ : and B Ÿ 2] Ê B−] 4] Ÿ B Ê B−] Hint: You may use an exercise from an earlier chapter, to wit: Let P be a lattice and let + − P. Let \ be a generating set for P. For any subset W of P, let T ÐWÑ be the property that for any finite subset W! © W , + Ÿ 2W! 8. Ê + = for some = − W! Show that if T Ð\Ñ holds then T ÐPÑ holds. Use this exercise to show that \ satisfies the conditions 1)–6) of Theorem 11.5. Then use a previous exercise from this chapter to finish this exercise. Show that the following hold in any free lattice P œ FLÐ\Ñ, for ?ß +ß , : SD” ) Ò?Ó œ Ò+Ó ” Ò,Ó and Ò?Ó œ Ò+Ó ” Ò-Ó Ê Ò?Ó œ Ò+Ó ” ÐÒ,Ó • Ò-ÓÑ SD• ) Ò?Ó œ Ò+Ó • Ò,Ó and Ò?Ó œ Ò+Ó • Ò-Ó Ê Ò?Ó œ Ò+Ó • ÐÒ,Ó ” Ò-ÓÑ These are referred to as the semidistributive laws. Free Lattices 9. 261 Let Ò>Ó − FLÐ\Ñ and let Ò?Ó − FLÐ\Ñ be join irreducible. Prove that Ò?Ó has a canonical form of the form Ò>Ó œ Ò?Ó ” Ò+" Ó ” â ” Ò+8 Ó if and only if there is an element Ò+Ó − FLÐ\Ñ for which Ò>Ó œ Ò?Ó ” Ò+Ó and if Ò@Ó Ò?Ó, then Ò@Ó ” Ò+Ó Ò>Ó. 10. Develop an algorithm for computing a canonical form for any Ò>Ó − FLÐ\Ñ. Chapter 12 Fixed-Point Theorems In this chapter, we give a brief introduction to the theory of fixed points for functions defined on a poset T . Let us begin with a few definitions. Definition Let T be a poset. A fixed point for a function 0 À T Ä T is an element + − T for which 0 Ð+Ñ œ + Let us also recall an earlier definition. Definition Let T be a poset. A function 0 À T Ä T is inflationary if B Ÿ 0 ÐBÑ for all B − T . The term increasing is also used for the previous concept but, unfortunately, some authors use increasing to mean monotone, so we have decided to avoid the term altogether. We also recall the following definition. Definition Let T and U be posets that have directed joins. A function 0 À T Ä U is (join) continuous if for any directed subset H in T , the set 0 ÐHÑ is also directed and Ä 0 Œ 2 .−H . Ä œ 2 .−H 0 Ð.Ñ If the function 0 À T Ä T and the poset T are sufficiently well behaved, then it is not hard to “construct” a fixed point for 0 . Take B − T and consider the sequence Bß 0 ÐBÑß 0 # ÐBÑß á S. Roman (ed.), Lattices and Ordered Sets, doi: 10.1007/978-0-387-78901-9_12, © Steven Roman 2008 263 264 Lattices and Ordered Sets If 0 is inflationary, then this is a chain in T . If T is chain-complete, then we may set α œ 10 8 ÐBÑ. If 0 is also continuous, then α Ÿ 0 ÐαÑ œ 0 Š20 8 ÐBÑ‹ œ 20 8" ÐBÑ Ÿ α and so α is a fixed point of 0 . Thus, if T is chain-complete and 0 is both inflationary and continuous, then 0 has a fixed point. Our goal in this chapter is to sharpen this result. Recall that a poset T is complete (also called a CPO) if it has a smallest element and has directed joins. Theorem 2.19 implies that a poset is complete if and only if it is chain-complete. Definition A sub-CPO of a CPO T is a subset W © T for which 1) !T − W 2) W inherits directed joins from T . Fixed Point Terminology Here are some terms associated with fixed points. Definition Let T be a poset and let 0 À T Ä T be a function. Let + − T . Then (see Figure 12.1) 1) + is a fixed point of 0 if 0 Ð+Ñ œ + 2) + is a pre-fixed point of 0 if 0 Ð+Ñ Ÿ + 3) + is a post-fixed point of 0 if 0 Ð+Ñ + The corresponding sets of fixed points are denoted by FixÐ0 Ñ, PreÐ0 Ñ and PostÐ0 Ñ. The smallest elements of these sets, if they exist, are denoted by MinFixÐ0 Ñ, MinPreÐ0 Ñ and MinPostÐ0 Ñ, respectively. The largest elements are denoted by MaxFixÐ0 Ñ, MaxPreÐ0 Ñ and MaxPostÐ0 Ñ. Fixed-Point Theorems 265 P fix(f) post(f) pre(f) P Figure 12.1 Recall that if 0 À W Ä W is a function, then a subset X © W is said to be 0 invariant if 0 ÐX Ñ © X . Theorem 12.1 Let 0 À T Ä T be a monotone map. 1) a) If T is a CPO, then PreÐ0 Ñ is 0 -invariant and inherits arbitrary nonempty existing T -meets. Moreover, MinPreÐ0 Ñ exists MinPreÐ0 Ñ œ MinFixÐ0 Ñ Ê b) If T is a complete lattice, then PreÐ0 Ñ is a meet-structure in P and so a complete lattice. 2) a) If T is a CPO, then PostÐ0 Ñ is 0 -invariant and inherits arbitrary nonempty existing T -joins. Moreover, MaxPostÐ0 Ñ exists MaxPostÐ0 Ñ œ MaxFixÐ0 Ñ Ê Since !T − PostÐ0 Ñ, it follows that PostÐ0 Ñ is a sub-CPO of T . b) If T is a complete lattice, then PostÐ0 Ñ is a join-structure in P and so a complete lattice. Proof. For part 1), the set PreÐ0 Ñ is 0 -invariant since 0 ÐBÑ Ÿ B If 3B3 exists for B3 − PreÐT Ñ, then Ê 0 Š4B3 ‹ Ÿ 0 ÐB3 Ñ Ÿ B3 0 Ð0 ÐBÑÑ Ÿ 0 ÐBÑ Ê 0 Š4B3 ‹ Ÿ 4B3 and so 3B3 − PreÐ0 Ñ. Finally, if 2 œ MinPreÐ0 Ñ then 0 Ð2Ñ − PreÐ0 Ñ implies that 0 Ð2Ñ Ÿ 2 Ÿ 0 Ð2Ñ and so 2 − FixÐ0 Ñ. Since FixÐ0 Ñ © PreÐ0 Ñ, it follows that 2 œ MinFixÐ0 Ñ. Part 2) is dual. Fixed-Point Theorems: Complete Lattices We are now ready to discuss theorems asserting the existence of fixed points, beginning with monotone maps on complete lattices. 266 Lattices and Ordered Sets Theorem 12.2 (The Knaster–Tarski Fixed Point Theorem, [39], [60]) Let P be a complete lattice and let 0 À P Ä P be a monotone map on P. Then FixÐPÑ is a complete lattice, with bounds MaxFixÐ0 Ñ œ MaxPostÐ0 Ñ œ 2PostÐ0 Ñ and MinFixÐ0 Ñ œ MinPreÐ0 Ñ œ 4PreÐ0 Ñ Proof. Note that FixÐ0 Ñ œ PreÐ0 Ñ ∩ PostÐ0 Ñ œ PostÐ0 lPreÐ0 Ñ Ñ But 0 lPreÐ0 Ñ is a monotone map on the complete lattice PreÐ0 Ñ and so PostÐ0 lPreÐ0 Ñ Ñ is also a complete lattice. The previous theorem has a converse, proved by Davis [13] in 1955. In particular, a lattice is complete if and only if every monotone function has a fixed point. Since the full proof requires the notions of ordinal and transfinite sequence, we give the proof first in the countable case. Theorem 12.3 A countable lattice P is complete if and only if every monotone function on P has a fixed point. Proof. We need only prove that if P is incomplete, then there is a monotone function with no fixed point. Let us recall Theorem 3.22: A countably infinite lattice P is incomplete if and only if it has a coalescing pair ÐFß GÑ of sequences. If P has an unbounded strictly increasing infinite sequence F œ Ð," ,# âÑ then for any B − P, there is a ,3 − F for which ,3 Ÿ y B. Let 0 ÐBÑ œ minÖ,3 ± ,3 Ÿ y B× Then 0 ÐBÑ Á B for all B − P, that is, 0 has no fixed point. Moreover, BŸC Ê Ö,3 ± ,3 Ÿ y C× © Ö,3 ± ,3 Ÿ y B× Ê 0 ÐBÑ Ÿ 0 ÐCÑ and so 0 is monotone. The case where P has a strictly decreasing infinite sequence G œ Ð-" -# âÑ with no lower bound is similar. In particular, for each B − P let 0 ÐBÑ œ maxÖ-3 ± B Ÿ y -3 × Then 0 ÐBÑ Á B for all B − P and Fixed-Point Theorems BŸC Ê Ö-3 ± B Ÿ y -3 × © Ö-3 ± C Ÿ y -3 × Ê 267 0 ÐBÑ Ÿ 0 ÐCÑ and so 0 is monotone. Finally, suppose that P has two strictly monotone coalescing infinite sequences F œ Ð," ,# âÑ and G œ Ðâ -# -" Ñ for which F G and F? ∩ G j œ g If B − G j , then B  F ? and we define 0 ÐBÑ by 0 ÐBÑ œ minÖ,3 ± ,3 Ÿ y B× On the other hand, if B  G j , then we let 0 ÐBÑ œ maxÖ-3 ± B Ÿ y -3 × In either case, 0 ÐBÑ Á B and we are left with showing that 0 is monotone. Let B C. If Bß C − G j or if Bß C  G j , then the same argument as above shows that 0 ÐBÑ Ÿ 0 ÐCÑ. The only other possibility is that B − G j and C  G j . Then 0 ÐBÑ œ minÖ,3 ± ,3 Ÿ y B× and 0 ÐCÑ œ maxÖ-3 ± C Ÿ y -3 × and since F G , it follows that 0 ÐBÑ Ÿ 0 ÐCÑ. Thus, 0 is monotone. Now we turn to the general case, whose proof is a direct generalization of the previous proof to the transfinite case. Theorem 12.4 (Davis, 1955 [13]) A lattice P is complete if and only if every monotone function on P has a fixed point. Proof. We need only prove that if P is incomplete, then there is a monotone function with no fixed point. Again, the proof uses Theorem 3.23. If P has a strictly increasing transfinite sequence U œ Ø,α ± α $ Ù with no upper bound, then for any B − P, the set of ordinals IB œ Öα $ ± ,α Ÿ y B× is nonempty and so has a least element -B . Let 0 ÐBÑ œ ,-B 268 Lattices and Ordered Sets Then -B − IB implies that 0 ÐBÑ Á B and so 0 has no fixed points. Moreover, if B Ÿ C, then BŸC Ê IC © I B Ê -B Ÿ -C Ê 0 ÐBÑ Ÿ 0 ÐCÑ and so 0 is monotone. The case where P has a strictly decreasing transfinite sequence V œ Ø-α ± α ,Ù with no lower bound is similar. In particular, for each B − P, the set of ordinals JB œ Ö α $ ± B Ÿ y -α × is nonempty and so has a least element .B . Let 0 ÐBÑ œ -.B Then 0 ÐBÑ Á B and BŸC Ê JB © JC Ê .C Ÿ .B Ê 0 ÐBÑ Ÿ 0 ÐCÑ and so 0 is monotone. Finally, suppose that P has two strictly monotone coalescing infinite sequences U œ Ø,α ± α $ Ù and V œ Ø-α ± α ,Ù for which U V and U ? ∩ Vj œ g If B − Vj , then B  U ? and we define 0 ÐBÑ by 0 ÐBÑ œ ,-B as above. On the other hand, if B  Vj , then we let 0 ÐBÑ œ -.B as above. In either case, 0 ÐBÑ Á B and we are left with showing that 0 is monotone. Let B C. If Bß C − Vj or if Bß C  Vj , then the same argument as above shows that 0 ÐBÑ Ÿ 0 ÐCÑ. The only other possibility is that B − Vj and C  Vj . Then 0 ÐBÑ œ ,-B − U and 0 ÐBÑ œ -.B − V However, since U V, it follows that 0 ÐBÑ Ÿ 0 ÐCÑ. Thus, 0 is monotone. Fixed-Point Theorems 269 Fixed–Point Theorems: Complete Posets We now turn to the question of the existence of fixed points for functions on a complete poset. As we will see, any inflationary or monotone function on a CPO T has a fixed point. However, we begin by considering functions that are both monotone and inflationary. Theorem 12.5 The set Y of all monotone, inflationary functions on a complete poset T has a common fixed point, that is, there is an + − T for which 0 Ð+Ñ œ + for all 0 − Y . Proof. The set T T of all functions on T is partially ordered by pointwise order: 0 Ÿ1 if 0 ÐBÑ Ÿ 1ÐBÑ for all B − T If 0 ß 1 − Y , then 0 ‰ 1 − Y , since BŸC Ê 1ÐBÑ Ÿ 1ÐCÑ Ê 0 Ð1ÐBÑÑ Ÿ 0 Ð1ÐCÑÑ and so 0 ‰ 1 is monotone. Also, B Ÿ 1ÐBÑ implies that B Ÿ 0 ÐBÑ Ÿ 0 Ð1ÐBÑÑ and so 0 ‰ 1 is inflationary. Thus, 0 ‰ 1 is an upper bound for 0 and 1 in Y and so Y is a directed family in T T . Hence, we can define a function 2À T Ä T by 2ÐBÑ œ 2Ö0 ÐBÑ ± 0 − Y × To see that 2 − Y , if B Ÿ C, then 0 ÐBÑ Ÿ 0 ÐCÑ for all 0 − Y and so 2ÐBÑ Ÿ 2ÐCÑ, which shows that 2 is monotone. Also, since B Ÿ 0 ÐBÑ for all 0 − Y , it follows that B Ÿ 2ÐBÑ and so 2 is inflationary. Thus, 2 is the largest element of Y . Finally, if 0 − Y , then 2 Ÿ 0 ‰ 2 Ÿ 2 , that is, 0 ‰ 2 œ 2 . It follows that for any + − T , we have 0 Ð2Ð+ÑÑ œ 2Ð+Ñ, that is, 2Ð+Ñ is a fixed point for every 0 − Y . To improve upon Theorem 12.5, the key is to consider the smallest 0 -invariant sub-CPO of T . In particular, let 0 À T Ä T be a function on a complete poset T and let Y œ ÖW © T ± W is an 0 -invariant sub-CPO of T × which is not empty since T − Y . Moreover, the intersection T! œ +Y is nonempty since ! − T! and T! − Y . Hence, T! is the smallest 0 -invariant subCPO of T . Monotone Functions on a CPO We can now show that any monotone function on a CPO has a fixed point. 270 Lattices and Ordered Sets Theorem 12.6 Let T be a CPO and let 0 À T Ä T be a monotone function. Then 0 has a least fixed point MinFixÐ0 Ñ. Proof. Since 0 is monotone, PostÐ0 Ñ − Y and since T! © PostÐ0 Ñ, it follows that 0 is inflationary on T! . Thus, the restriction of 0 to T! is an inflationary monotone function on T! and so Theorem 12.5 implies that 0 has a fixed point α − T! . Moreover, if + − FixÐ0 Ñ, then Æ + − Y since B − Æ + Ê B Ÿ + Ê 0 ÐBÑ Ÿ 0 Ð+Ñ œ + Ê 0 ÐBÑ − Æ + and since Æ + inherits directed joins from T . Hence, T! © Æ +, that is, T! Ÿ +. It follows that there is exactly one fixed point α − T! and α œ MinFixÐ0 Ñ. Inflationary Functions on a CPO We now turn to inflationary functions. Theorem 12.7 Let T be a CPO and let 0 À T Ä T be an inflationary function. Then 0 has a fixed point. Proof. We again consider the smallest 0 -invariant sub-CPO T! of T . Our goal is to show that T! is directed, since then the join α œ 1T! exists and so α Ÿ 0 Ð α Ñ Ÿ 2T ! œ α which shows that α − FixÐ0 Ñ. If we show that for all + − T! , every element , − T! satisfies , Ÿ + or , 0 Ð+Ñ, then T! is actually a chain, for in this case, we have either , Ÿ + or , 0 Ð+Ñ + and so + and , are comparable. Thus, we wish to show that T! œ ^+ ³ Ò!ß +Ó ∪ Ò0 Ð+Ñß ∞Ñ where the intervals are taken with respect to T! . Since ^+ © T! , we need only show that ^+ is an 0 -invariant sub-CPO of T , since then T! © ^+ . It is easy to see that ^+ is a sub-CPO of T . Clearly, ! − ^+ . Also, if H œ Ö.3 × © ^+ is directed, let . œ 1T H − T! . If .3 Ÿ + for all 3, then . Ÿ + and so . − ^+ and if 0 Ð+Ñ Ÿ .3 for some 3, then 0 Ð+Ñ Ÿ .3 Ÿ . and so . − ^+ . Thus ^+ is a sub-CPO of T for any + − T! . As to the 0 -invariance of ^+ , since 0 is inflationary, 0 maps the interval Ò0 Ð+Ñß ∞Ñ into itself. Also, 0 maps + to 0 Ð+Ñ − ^+ . Finally, we show that 0 sends Ò!ß +Ñ into itself, that is, , − T! ß , + Ê 0 Ð,Ñ Ÿ + Let e be the set of all + − T! for which this is true, that is, e œ Ö+ − T! ± , − T! ß , + Ê 0 Ð,Ñ Ÿ +× Thus, + − e implies that ^+ œ T! . To see that e œ T! , it is sufficient to show that e is an 0 -invariant sub-CPO of T . Fixed-Point Theorems 271 It is clear that ! − e. For 0 -invariance, we must show that + − e implies , − T! œ ^+ ß , 0 Ð+Ñ Ê 0 Ð,Ñ Ÿ 0 Ð+Ñ If , +, then + − e implies that 0 Ð,Ñ Ÿ + Ÿ 0 Ð+Ñ; if , œ +, then clearly 0 Ð,Ñ Ÿ 0 Ð+Ñ; the case , 0 Ð+Ñ is not relevant. To show that e inherits directed joins, let H œ Ö+3 × © e be directed. We must show that , 2 +3 0 Ð,Ñ Ÿ 2+3 Ê Since , − ^+3 for each 3, it follows that , +3 or , œ +3 or , 0 Ð+3 Ñ +3 If , +5 for some 5 , then 0 Ð,Ñ Ÿ +5 Ÿ 1+3 , as desired, so we may assume that , y +3 for all 3. If , 0 Ð+3 Ñ +3 for all 3, then , 1+3 which is not the case. Hence, there is an index 5 for which , œ +5 . Since , y +3 for all 3, it follows that +5 œ , is maximal in H. But if +3 ² +5 œ , for some 3, then since H is directed, there is an +4 +5 , which is false. Hence, +5 œ , is the greatest element of H and so 1+3 œ +5 œ , , which is also false. These contradictions imply that , +5 for some 5 and complete the proof. Constructing a Fixed Point A first attempt to construct a fixed point might be to look at the sequence !ß 0 Ð!Ñß 0 # Ð!Ñß á Indeed, if a function 0 À T Ä T on a complete poset T is either inflationary or monotone, then this sequence is increasing. This is clear when 0 is inflationary and when 0 is monotone, we have ! Ÿ 0 Ð!Ñ and applying 0 8 gives 0 8 Ð!Ñ Ÿ 0 8" Ð!Ñ. Hence, the join Ä α œ 2 0 8 Ð!Ñ 8 ! exists. Now, if 0 is continuous, then 0 Ðα Ñ œ 0 2 0 8 Ð!Ñ 8 ! œ 2 0 8" Ð!Ñ œ 2 0 8 Ð!Ñ œ α 8 ! 8 ! and so α is a fixed point. In addition, if 0 is monotone and + is any fixed point of 0 , then ! Ÿ + implies that 0 8 Ð!Ñ Ÿ 0 8 Ð+Ñ œ + and so α Ÿ +, that is, α œ MinFixÐ0 Ñ 272 Lattices and Ordered Sets Theorem 12.8 Let 0 À T Ä T be a continuous function on a complete poset T . If 0 is either monotone or inflationary, then 0 has a fixed point Ä α œ 2 0 8 Ð!Ñ 8 ! and if 0 is monotone, then α œ MinFixÐ0 Ñ With the help of ordinals, we can generalize Theorem 12.8 to perform a transfinite “construction” of a fixed point that works for both monotone and inflationary functions on a CPO, without requiring continuity. A transfinite sequence Ø+α ± α − ordÙ is increasing if α " implies +α Ÿ +" and strictly increasing if α " implies +α +" . Let T be a CPO and let 0 À T Ä T be either a monotone or an inflationary function. Consider the following proposition T ÐαÑ for α − ord: 1) If α œ ! and +α œ !, then the sequence Tα œ Ø+# ± # Ÿ αÙ is increasing. 2) If α œ " " for " − ord and +α œ 0 Ð+" Ñ, then the sequence Tα œ Ø+# ± # Ÿ αÙ is increasing. 3) If α is a limit ordinal, the sequence Uα œ Ø+# ± # αÙ is increasing and if +α œ 1Uα , then the sequence Tα œ Ø+# ± # Ÿ αÙ is also increasing. We prove that T ÐαÑ holds by induction on α. If α œ !, the result is clear. Suppose the result holds for all # α. If α œ " ", then T Ð" Ñ holds and so the sequence T" œ Ø+# ± # Ÿ " Ù œ Ø+# ± # αÙ is increasing and we want to show that Tα œ Ø+# ± # Ÿ αÙ œ T" ∪ Ö+α × is increasing and so it is sufficient to show that +" Ÿ +α , since +" is the largest element of T" . If 0 is inflationary, then +" Ÿ 0 Ð+" Ñ œ +" " œ +α If 0 is monotone, we proceed as follows. If " œ . ", then +. ß +" − T" and so +. Ÿ +" . Applying 0 gives +" œ +." œ 0 Ð+. Ñ Ÿ 0 Ð+" Ñ œ +" " œ +α If " is a limit ordinal, then for any # " , we have +# Ÿ +" and so Fixed-Point Theorems 273 +#" œ 0 Ð+# Ñ Ÿ 0 Ð+" Ñ œ +" " whence 2Ö+#" ± # " × Ÿ +" " œ +α and since +! œ !, the join on the left is +" . Thus, +" Ÿ +α in both cases and so Tα is increasing. Finally, suppose that α is a limit ordinal. Then T Ð" Ñ holds for all " α. To see that Uα is increasing, let # " α. Then T Ð" Ñ implies that the sequence T" is increasing and so +# Ÿ +" . Hence, Uα is increasing. To see that Tα is increasing, if " α, then +α œ 2Uα +" Thus, we have shown that T ÐαÑ holds for all α − ord. Theorem 12.9 Let T be a CPO and let 0 À T Ä T be either a monotone or an inflationary function. Then 0 has a fixed point. Moreover, the transfinite sequence T œ Ø+α ± α − ordÙ defined by +! œ ! +α" œ 0 Ð+α Ñ for all α − ord +- œ supÖ+" ± " -× for - a limit ordinal is well defined and increasing. 1) The sequence T eventually becomes constant, that is, there is an ordinal $ for which +$ œ +α for all α $ . Moreover, +$ is the only fixed point of 0 contained in T. 2) If 0 is monotone, then +$ is the least fixed point MinFixÐ0 Ñ of 0 . Proof. We have seen that the sequence T is well defined, the only issue being that the set U- œ Ö+" ± " -× is directed, and that T is increasing, since if " α, then +" and +α both lie in the increasing sequence Tα © T. However, T cannot be strictly increasing, since then kT k k" k for all ordinals " , which is not possible. Hence, there are ordinals α " for which +α œ +" and so the values of +# are equal for all α Ÿ # Ÿ " . It follows that T contains a fixed point of 0 , for if " is a limit ordinal, then α α " " and so +α œ +α" œ 0 Ð+α Ñ and if " œ # ", then +# œ +#" œ 0 Ð+# Ñ In either case, 0 has a fixed point +$ . Moreover, an induction argument shows that +α œ +$ for all α $ . Assume that +" œ +$ for all $ Ÿ " α. If 274 Lattices and Ordered Sets α œ " ", then +" œ +$ and so +α œ 0 Ð+" Ñ œ 0 Ð+$ Ñ œ +$ and if α is a limit ordinal, then +α œ 2Ö+" ± " α× œ +$ It follows that T contains exactly one fixed point of 0 and 1) holds. For part 2), if 0 is monotone, we show by induction that any fixed point B of 0 satisfies T Ÿ B. Clearly, ! Ÿ B. If +α Ÿ B, then +α" œ 0 Ð+α Ñ Ÿ 0 ÐBÑ œ B and so +α" Ÿ B. Also, if - is a limit ordinal and +α Ÿ B for all α -, then +- œ 2Ö+α ± α -× Ÿ B Thus, +$ − T is the least fixed point of 0 . Exercises 1. Let T be a CPO. Consider the sets infmonÐT Ñ © monÐT Ñ © funcÐT Ñ of inflationary, monotone functions on T , monotone functions on T and all functions on T , respectively, under the pointwise partial order 0 Ÿ1 a) if 0 ÐBÑ Ÿ 1ÐBÑ for all B − T Prove that funcÐT Ñ is a CPO, where Ä Ä Œ 2 03 ÐBÑ œ 2 03 ÐBÑ 2. 3. 4. 5. 6. 7. b) Prove that monÐT Ñ is a sub-CPO of funcÐT Ñ. Show that a nonidentity inflationary function 0 on a CPO T need not have a least fixed point. Show that an inflationary, monotone function on a poset T need not have a fixed point. Prove that any poset with ! and with the ACC is a CPO. Let T and U be CPOs and let 0 À T Ä U be monotone. Prove that if T has the ACC then 0 is continuous. Find an example of a CPO T and a nonempty subset W © T that is a CPO under the same order but is not a sub-CPO of T . Hint: Look at a special family of subsets of the natural numbers . Let \ be a nonempty set and let \ ‡‡ be the set of all finite or infinite strings over the alphabet \ . Order \ ‡‡ by saying that the empty word is the smallest element and that A Ÿ @ if A is a prefix of @, that is, if @ œ A? for Fixed-Point Theorems 275 some string ?. Show that \ ‡‡ is a CPO under this order. What about the set \ ‡ of finite strings over \ ? 8. Let T be a countable poset. Prove that T is complete if and only if it is completely inductive. 9. Prove that the underlying set of the sequence T from Theorem 12.9 is the set T! . Hence, there is a unique fixed point of 0 in the set T! , even when 0 is inflationary. However, show that it need not be the least fixed point of 0 in T . 10. A subset W of a poset T is consistent if every pair of elements of W has an upper bound in T (but not necessarily in W ). Let T be a CPO. Prove that the following are equivalent: a) T is consistently complete, that is, every consistent subset of T has a join in T . b) All subsets that are bounded from above (have an upper bound in T ) have a join in T . c) All nonempty subsets of T have a meet in T . d) T ð (T with a largest element adjoined) is a complete lattice. e) Æ B is a complete lattice for every B − T . 11. Let T be a poset and let Ö+3ß4 ± 3 − Mß 4 − N × be a doubly-indexed subset of T. a) Show that if for all ? − M and @ − N , the joins 2 +?ß4 , 4 2 +3ß@ 3 and 2 Š2 +3ß4 ‹ 3 4 exist then 13ß4 +3ß4 and 14 13 +3ß4 exist and 2 +3ß4 œ 2 Š2 +3ß4 ‹ œ 2 Š2 +3ß4 ‹ 3ß4 3 4 4 3 b) Let T and U be posets and let Y ÐT ß UÑ be the poset of functions from T to U, with pointwise ordering: 0 Ÿ 1 if 0 ÐBÑ Ÿ 1ÐBÑ for all B − T . Show that if U is a CPO then so is Y ÐT ß UÑ. c) Let T be a poset and let U be a CPO. Let VÐT ß UÑ be the poset of continuous functions from T to U, with pointwise ordering. Show that VÐT ß UÑ is a sub-CPO of Y ÐT ß UÑ. 12. Show that the KnasterTarski theorem can be used to prove the SchröderBernstein theorem: If E and F are sets and lEl Ÿ lFl and lF l Ÿ lEl then lEl œ lFl. Hint: To prove the SchröderBernstein theorem, show that if 0 À E Ä F and 1À F Ä E are injective maps then there is a bijection between E and F . Prove that it is sufficient to find subsets W © E and X © F such that ÖWß 1ÐX Ñ× is a partition of E and ÖX ß 0 ÐWÑ× is a partition of F . Show that this is equivalent to the existence of a subset W © E such that W œ E Ï 1ÐF Ï 0 ÐWÑÑ Appendix A1 A Bit of Topology We present the topology needed for the discussion of the representation theorem for distributive lattices. Topological Spaces Definition Let \ be a nonempty set. A topology on \ is a nonempty family 7 of subsets of \ with the following properties: 1) gß \ − 7 2) 7 is closed under arbitrary unions 3) 7 is closed under finite intersections. If 7 is a topology on \ , then Ð\ß 7 Ñ is a topological space. One often says that \ is a topological space when the topology is understood (or does not need explicit mention). The elements of 7 are called open sets. The complement of an open set is a closed set. A subset that is both open and closed is clopen. Definition If \ is a nonempty set, then the power set kÐ\Ñ is a topology on \ , called the discrete topology. In a topology, every subset of \ is open as well as closed and clopen. Subspaces Theorem A1.1 Let Ð\ß 7 Ñ be a topological space. Let ] be a nonempty subset of \ . The family 5 œ ÖE ∩ ] ± E − 7 × is a topology on ] , called the induced topology, and the topological space Ð] ß 5Ñ is called a subspace of Ð\ß 7 Ñ. We usually say that ] is a subspace of Ð\ß 7 Ñ. Bases and Subbases One way to describe the open sets of a topological space is to give a basis or subbasis for it, as described in the following definition. 278 Lattices and Ordered Sets Definition Let Ð\ß 7 Ñ be a topological space. 1) A family U of open sets in 7 is said to be a basis (or base) for 7 if every open set in 7 is the union of elements of U . 2) A family f of open sets in 7 is said to be a subbasis (or subbase) for 7 if every open set in 7 is a union of finite intersections of elements of f . Note that the union of an empty family is the empty set and the intersection of an empty family is the entire space. Any nonempty family Y of subsets of a set \ qualifies to be a subbasis for a topology: Just take the collection of all unions of finite intersections of members of Y . On the other hand, not every nonempty family of subsets of \ qualifies as a basis for a topology. The following theorem characterizes those families of subsets that do qualify. Theorem A1.2 Let \ be a nonempty set and let Y be a nonempty family of subsets of \ . Then Y ∪ Ögß \× is a basis for a topology on \ if and only if the intersection of any two members of Y is a union of members of Y . Connectedness and Separation Definition Let Ð\ß 7 Ñ be a topological space. A separation in Ð\ß 7 Ñ is a partition ÖY ß Z × of \ by open sets. (Hence, Y and Z are nonempty.) If ? − Y and @ − Z we will say that ÐY ß Z Ñ is a separation of Ð?ß @Ñ. Note that if ÖY ß Z × is a separation, then Z œ Y - and so Y is a nonempty proper clopen set. Conversely, a nonempty proper clopen set G defines a separation ÖGß G - ×. Definition Let Ð\ß 7 Ñ be a topological space. 1) Ð\ß 7 Ñ is Hausdorff if for any pair B Á C − \ there are disjoint open sets Y and Z such that B − Y and C − Z . 2) Ð\ß 7 Ñ is connected if \ does not have a separation or equivalently, if the only clopen sets in \ are g and \ . 3) Ð\ß 7 Ñ is totally separated if every pair of distinct elements has a separation or equivalently, if for every B Á C − \ there is a clopen set G such that B − G and C  G . Compactness Definition Let Ð\ß 7 Ñ be a topological space. 1) An open cover of \ is a family h of open sets in 7 whose union is \ . If i © h is also an open cover of \ , then i is a subcover of h . 2) Ð\ß 7 Ñ is compact if every open cover of \ has a finite subcover. Theorem A1.3 Let Ð\ß 7 Ñ be a topological space. Appendix A1: A Bit of Topology 279 1) Ð\ß 7 Ñ is compact if and only if any open cover of \ by basis elements has a finite subcover. 2) A closed subset of a compact space is compact (in the induced topology). We will have occasion to use the following result, whose proof seems a bit scarce, so we present it here. Theorem A1.4 (Alexander subbasis lemma) Let Ð\ß 7 Ñ be a topological space with subbasis f . If every cover of \ by elements of f has a finite subcover then \ is compact. Proof. Let … be the collection of all open covers of \ that do not have finite subcovers. For the purposes of contradiction, assume that … is nonempty and order … by set inclusion. We wish to show that … has a maximal element. Let ‚ be a chain in … and let h œ -‚. If h has a finite subcover ^ œ ÖS" ß á ß S8 ×, then there is a V − ‚ that contains each S3 and so V has the finite subcover ^ as well, which is a contradiction. Hence, h has no finite subcover and so h − … is an upper bound for the chain V. Thus, Zorn's lemma implies that there is a maximal open cover ` of \ having no finite subcover. The cover ` has some special properties. First, if S  ` is open, then the maximality of ` implies that the open cover ` ∪ ÖS× has a finite subcover and so \ œ S ∪ Q" ∪ â ∪ Q 5 for some Q3 − `. It follows that no open superset of S can be in `. Thus, if E − `, then all open subsets S of E are in ` as well. Second, if E and F are open and E  ` and F  `, then E ∪ Q" ∪ â ∪ Q 5 œ \ œ F ∪ R " ∪ â ∪ R < for some R3 ß Q3 − ` and so \ œ ÐE ∩ FÑ ∪ Q" ∪ â ∪ Q5 ∪ R" ∪ â ∪ R< which implies that E ∩ F  `. In other words, E ∩ F − ` implies that E − ` or F − `. This extends to any finite intersection of open sets as well. Now consider the intersection a œ ` ∩ f , that is, the elements of ` that are also subbasis elements. We claim that a is also a cover for \ , which cannot be, since a has a finite subcover, which would also be a finite subcover consisting of elements of `. If B − \ , then there is an Q − ` with B − Q . Since Q is open, there is a basis element F for which B − F © Q . Thus, we have seen that F − `. Also, F œ W" ∩ â ∩ W: , for some W3 − f and so 280 Lattices and Ordered Sets B − F œ W" ∩ â ∩ W: Hence, by the preceding discussion, W3 − ` for some 3 and so W3 − a . Thus, a covers \ . Continuity Definition Let Ð\ß 7 Ñ and Ð] ß 5Ñ be topological spaces. A function 0 À \ Ä ] is continuous if the inverse image of every open set in ] is open in \ , that is, Y −5 Ê 0 " Y − 7 A continuous bijection 0 whose inverse 0 " is also continuous is called a homeomorphism. Theorem A1.5 Let Ð\ß 7 Ñ and Ð] ß 5Ñ be topological spaces. The following are equivalent for a function 0 À \ Ä ] : 1) 0 is continuous. 2) The inverse image of any closed set is closed. 3) If U is a basis for ] , then the inverse image of any element of U is open in \. 4) If f is a subbasis for ] , then the inverse image of any element of f is open in \ . Theorem A1.6 Let Ð\ß 7 Ñ and Ð] ß 5Ñ be topological spaces and let 0 À \ Ä ] be continuous. 1) If G is compact in \ then 0 ÐGÑ is compact in ] . 2) If G is connected in \ then 0 ÐGÑ is connected in ] . 3) If \ is compact Hausdorff and ] is Hausdorff, then 0 is a homeomorphism if and only if it is a bijection. We have remarked that closed subsets of a compact space are compact. In a compact Hausdorff space, the converse is also true. Theorem A1.7 In a compact Hausdorff space, the compact subsets are the same as the closed subsets. The Product Topology If k œ ÖÐ\3 ß 73 Ñ ± 3 − M× is a family of topological spaces, then the cartesian product \ œ $3−M \3 œ Ö0 À M Ä .3−M \3 ± 0 Ð3Ñ − \3 × can be given a topology 7 called the product topology as follows. For each 3 − M , let 13 À \ Ä \3 be the projection map, defined by 13 Ð0 Ñ œ 0 Ð3Ñ for all 0 − \ . For F3 © \3 , the set Appendix A1: A Bit of Topology 281 13" ÐF3 Ñ œ Ö0 − \ ± 0 Ð3Ñ − F3 × is called a cylinder with base F3 . It is the set of all functions that map the index 3 into the set F3 . A cylinder is called an open cylinder if its base is open and a closed cylinder if its base is closed. Note that the entire space \ is a cylinder. All other cylinders are proper cylinders. A subbasis for the product topology 7 consists of all open cylinders 13" ÐFÑ. Thus, a base for 7 consists of g, \ and all sets of the form Y3" ßáß38 œ 13" ÐF3" Ñ ∩ â ∩ 13" ÐF38 Ñ " 8 œ Ö0 − \ ± 0 Ð35 Ñ − F35 for 5 œ "ß á ß 8× where F35 is open in \35 and 35 − M . In words, the basic open sets are the functions whose values on a finite set Ö3" ß á ß 38 × of indices must lie in the prescribed open sets F3" ß á ß F38 . It follows that a proper subset S of \ is open in the product topology on \ if and only if S is the union of sets from Y3" ßáß38 . It is easy to see that the projection maps are continuous and surjective. Moreover, 13 is also an open map, that is, 13 maps open sets in \ to open sets in \3 . To see this, note that 13 sends any basic open set Y3" ßáß38 in \ to an open set in \3 , namely, \3 if 3  Ö3" ß á ß 38 × or F35 if 3 œ 35 . Also, 13 preserves union. It follows that a cylinder is open in the product topology if and only if it has an open base. Moreover, since 13" is a Boolean homomorphism, a cylinder is closed in the product topology if and only if its base is closed. Thus, the terms open cylinder and closed cylinder are unambiguous. Appendix A2 A Bit of Category Theory Here we present the category theory that we will need in the text. Categories Informally speaking, a category consists simply of a class of mathematical objects and their structure-preserving maps. Definition A category V consists of the following: 1) (Objects) A class ObjÐVÑ whose elements are called the objects of the category. 2) (Morphisms) For each pair ÐEß FÑ of objects of V, a set homV ÐEß FÑ whose elements are called morphisms, maps or arrows from E to F . If 0 − homV ÐEß FÑ we write 0 À E Ä F . 3) The sets homV ÐEß FÑ are disjoint for distinct pairs ÐEß FÑ. 4) (Composition) For 0 À E Ä F and 1À F Ä G , there is a morphism 1 ‰ 0 À E Ä G , called the composition of 1 with 0 . Moreover, composition is associative: 0 ‰ Ð1 ‰ 2Ñ œ Ð0 ‰ 1Ñ ‰ 2 whenever the compositions are defined. 5) (Identity morphisms) For each object E − ObjÐVÑ there is a morphism +E À E Ä E called the identity morphism for E, with the property that if 0 À E Ä F, then +F ‰ 0 œ 0 and 0 ‰ +E œ 0 One often writes E − V as a shorthand for E − ObjÐVÑ. Definition A morphism 0 À E Ä F is an isomorphism if there is a morphism 0 " À F Ä E, called the (two-sided) inverse of 0 , for which 0 " ‰ 0 œ +E and 0 ‰ 0 " œ +F In this case, the objects E and F are isomorphic and we write E ¸ F . 284 Lattices and Ordered Sets Example A2.1 Here are some examples of common categories. In most cases, composition is the “obvious” one. The Category Set of Sets Obj is the class of all sets. homÐEß FÑ is the set of all functions from E to F . The Category Grp of Groups Obj is the class of all groups. homÐEß FÑ is the set of all group homomorphisms from E to F . The Category Ab of Abelian Groups Obj is the class of all abelian groups. homÐEß FÑ is the set of all group homomorphisms from E to F . The Category VectJ of Vector Spaces over a Field J Obj is the class of all vector spaces over J . homÐEß FÑ is the set of all linear transformations from E to F . The Category Rng of Rings Obj is the class of all rings (with unit). homÐEß FÑ is the set of all ring homomorphisms from E to F . The Category Poset of Partially Ordered Sets Obj is the class of all partially ordered sets. homÐEß FÑ is the set of all monotone functions from E to F . The Category FinDLat of all finite distributive lattices Obj is the class of all finite distributive lattices. homÐEß FÑ is the set of all lattice Ö!ß "×-homomorphisms from E to F . The Category Top of Topological Spaces Obj is the class of all topological spaces. homÐEß FÑ is the set of all continuous functions from E to F . Here are a couple of examples of more unusual categories. The Category MatrJ of Matrices Obj is the set of all positive integers. homÐ-ß <Ñ is the set of all < ‚ - matrices over J , composition being matrix multiplication (which accounts for the order of < and - in homÐ-ß <Ñ). The Category PosetÐT ß Ÿ Ñ, where ÐT ß Ÿ Ñ is a partially ordered set. Obj is the set of elements of c . Appendix A2: A Bit of Category Theory 285 homÐEß FÑ is empty unless E Ÿ F , in which case homÐEß FÑ contains a single element, denoted by EF . Note that the hom-sets specify the relation Ÿ on T . As to composition, there is really only one choice: If EFÀ E Ä F and FGÀ F Ä G then it follows that E Ÿ F Ÿ G and so E Ÿ G , which implies that homÐEß GÑ Á g. Thus, we set FG ‰ EF œ EG . The hom-set homÐEß EÑ is nonempty and contains only the identity morphism for E. Subcategories are defined as follows. Definition A category W is a subcategory of a category V if 1) ObjÐWÑ is a subclass of ObjÐVÑ 2) For every Eß F − ObjÐWÑ homW ÐEß FÑ © homV ÐEß FÑ 3) If 0 À E Ä F and 1À F Ä G are morphisms in W, then the composition 1 ‰ 0 taken in V is the composition of W. 4) The identity maps in V are in morphisms in W. Functors Structure-preserving maps between categories are called functors. Since a category consists of both objects and morphisms, a functor must consist of two functions—one to handle the objects and one to handle the morphisms. Also, there are two versions of functors: covariant and contravariant. Definition Let V and W be categories. A functor J À V Ê W is a pair of functions (as is customary, we use the same symbol J for both functions): 1) An object function J À ObjÐVÑ Ä ObjÐWÑ that maps objects in V to objects in W. 2) For a covariant functor, a morphism function J À MorÐVÑ Ä MorÐWÑ that maps each morphism 0 À E Ä F in V to a morphism J 0 À J E Ä J F in W. For a contravariant functor, a morphism function J À MorÐVÑ Ä MorÐWÑ that maps each morphism 0 À E Ä F in V to a morphism J 0 À J F Ä J E in W (note the reversal of direction). 3) Moreover, identity and composition are preserved, that is, for a covariant functor, J +E œ +J E and J Ð1 ‰ 0 Ñ œ J 1 ‰ J 0 and for a contravariant functor, 286 Lattices and Ordered Sets and J +E œ +J E J Ð1 ‰ 0 Ñ œ J 0 ‰ J 1 whenever all compositions are defined. The notation J À V Ê W is read “J is a functor from V to W.” A functor J À V Ê V from V to itself is referred to as a functor on V . One informal way to think of a covariant functor J À V Ê W is as a mapping of one-arrow diagrams in V 0 EÒF to one-arrow diagrams in W J0 JE Ò JF with the property that loops and triangles are preserved, as shown in Figure A2.1. f A F ιΑ A B ιFA=FιΑ FA F g gðf FA Ff FgðFf=F(gðf) C FB Fg FC Figure A2.1 Similar pictures can be drawn for contravariant functors. Functors can be composed in the “obvious” way. Specifically, if J À V Ê W and KÀ W Ê X are functors, then K ‰ J À V Ê X is defined by ÐK ‰ J ÑÐEÑ œ KÐJ EÑ for E − ObjÐVÑ and ÐK ‰ J ÑÐ0 Ñ œ KÐJ 0 Ñ for 0 − homV ÐEß FÑ. Definition Let J À V Ê W be a functor. 1) J is full if for every pair Eß F of objects in V , the morphism map is surjective. 2) J is faithful if for every pair Eß F of objects in V , the morphism map is injective. Appendix A2: A Bit of Category Theory 287 Examples of Functors We give two simple examples of functors. Example A2.2 The power set functor cÀ Set Ê Set sends a set E to its power set kÐEÑ and each set function 0 À E Ä F to the induced function c0 À kÐEÑ Ä kÐFÑ that sends a subset \ © E to 0 \ . It is easy to see that this defines a covariant functor. Example A2.3 The following situation is quite common. Let V be a category. Suppose that W is another category with the property that every object in V can be thought of as an object in W and every arrow 0 À E Ä F of V can be thought of as an arrow 0 À E Ä F of W. For instance, every object in Grp can be thought of as an object in Set simply by ignoring the group operation and every group homomorphism between groups can be thought of as a set function between the corresponding sets. Similarly, every ring can be thought of as an abelian group by ignoring the ring multiplication. Functors such as these that “forget” some structure are called forgetful functors. For categories whose objects are algebraic structures, the “most forgetful” functor is the one that forgets all algebraic structure and thinks of an object (group, ring, field, poset, lattice, etc.) simply as a set. Example A2.4 The spectrum functor J À FinDLat Ê FinPoset sends a finite distributive lattice P to its spectrum SpecÐPÑ and a lattice homomorphism 0 À P Ä Q to the induced inverse map 0 " À SpecÐQ Ñ Ä SpecÐPÑ, restricted to prime ideals. In the opposite direction, the down-set functor bÀ FinPoset Ê FinDLat sends a finite poset T to the lattice bÐT Ñ of downsets of T and a monotone map 1À T Ä U between posets to the induced inverse map 1" À bÐUÑ Ä bÐT Ñ, restricted to down-sets. Natural Transformations A structure-preserving map between functors is called a natural transformation. Consider a pair of covariant functors J ß KÀ V Ê W, as shown in Figure A2.2. FA F A f Ff λA B G FB λB GA Figure A2.2 Gf GB 288 Lattices and Ordered Sets Intuitively speaking, the essence of a functor J is how it maps one-arrow † K is intended to map, diagrams 0 À E Ä F and a natural transformation -À J Ä in a nice way, the family of one-arrow diagrams that characterize the functor J to the corresponding family of one-arrow diagrams that characterize K. As shown in Figure A2.2, this is accomplished by a family Ö-E À J E Ä KE ± E − ObjÐVÑ× of morphisms in W—one for each object in V— with the property that the square in Figure A2.2 commutes, that is, K0 ‰ -E œ -F ‰ J 0 (We also use the notation -ÐEÑ in place of -E when that is more convenient.) Let us now give the formal definition of natural transformation. Definition Let J ß KÀ V Ê W be covariant functors. A natural transformation - from J to K is a family - œ Ö-E À J E Ä KE ± E − ObjÐVÑ× of morphisms in W, indexed by the objects in V, for which the square in Figure A2.2 commutes, that is, -F ‰ J 0 œ K0 ‰ -E † K or Ö-E ×À J Ä † K. Each for any 0 À E Ä F . This is denoted by -À J Ä morphism -E is referred to as a component of - . If each component -E is an isomorphism, then - is called a natural isomorphism and we write † Ö-E ×À J ¸ K. † " If -À J ¸ K is a natural isomorphism from J to K, then the family Ö-E × is a natural isomorphism from K to J . Let us consider a few examples of natural transformations. Example A2.5 (The determinant) Define functors J ß KÀ CRng Ê Grp as follows. The functor J maps a ring V to the group KP8 ÐVÑ of nonsingular matrices over V . If 0 À V Ä W is a ring homomorphism, then ÐJ 0 ÑQ is the matrix formed from Q by applying 0 to each entry of Q . The functor K maps a ring V to the group V ‡ of units of V and K0 is just the restriction of 0 to V ‡ . Now, the deteminant detV is a group homomorphism from KP8 ÐVÑ to V ‡ . As shown in Figure A2.3, Appendix A2: A Bit of Category Theory GLn(R) F f R Ff GLn(S) detR S G 289 detS R* Gf S* Figure A2.3 the condition that the determinant be natural is -ÐWÑ ‰ J 0 œ K0 ‰ -ÐVÑ that is, Ô 0 +"ß" det Õ 0 +8ß" â 0 +"ß8 × Î Ô +"ß" ã œ 0 det Ï Õ +8ß" â 0 +8ß8 Ø â +"ß8 ×Ñ ã â +8ß8 ØÒ for any matrix E œ Ð+3ß4 Ñ in KP8 ÐVÑ. Since this holds, the determinant is a natural transformation from J to K. Example A2.6 (The coordinate map) Let Vect‡5 be the category of nonzero vector spaces over the field 5 . To denote the fact that a vector space Z has dimension 8, we write Z8 . Fix an ordered basis for each vector space Z . Let >À Vect‡5 Ê Vect5‡ be the matrix representation functor that sends Z8 to the vector space 5 8 and a linear map 7 À Z8 Ä [7 to multiplication .Ò7 Ó by the matrix representation Ò7 Ó of 7 with respect to the chosen bases for Z and [ . Thus, .Ò7 Ó ÐBÑ œ .Ò7 Ó B for all B − 5 8 . The coordinate map 9Z À Z8 Ä 5 8 , which sends a vector @ − Z to its coordinate matrix Ò@Ó with respect to the chosen ordered basis for Z8 , is an isomorphism of vector spaces. Moreover, the family Ö9Z × is a natural isomorphism from the identity functor M to >. In fact, as shown in Figure A2.4, Vn I Vn τ τ Wm φV Wm Γ Figure A2.4 φW kn µ[τ] km 290 Lattices and Ordered Sets the well-known formula Ò7 @Ó œ Ò7 ÓÒ@Ó for 7À Z Ä [ is precisely the condition of naturalness Ð9[ ‰ 7 Ñ@ œ ÐÒ7 Ó ‰ 9Z ÑÐ@Ñ that is, 9[ ‰ 7 œ Ò7 Ó ‰ 9Z Example A2.7 (The double-dual map) Let FinVect be the category of finitedimensional vector spaces over a field 5 . Recall that the natural map -ÐZ ÑÀ Z Ä Z ‡‡ from a vector space Z to its second dual Z ‡‡ is defined by -ÐZ ÑÐ@Ñ œ s@ where s@ is evaluation at @, that is, s@Ð0 Ñ œ 0 Ð@Ñ. The natural map is always injective but it is surjective if and only if Z is finite-dimensional. Now, the operator adjoint 7 ‚ À [ ‡ Ä Z ‡ of a linear map 7 À Z Ä [ is defined by 7 ‚ Ð0 Ñ œ 0 ‰ 7 Hence, the second operator adjoint 7 ‚‚ À Z ‡‡ Ä [ ‡‡ is defined by 7 ‚‚ ÐαÑ œ α ‰ 7 ‚ for α − Z ‡‡ . Since every α − Z ‡‡ has the form α œ s@ for some @ − Z , this can be written 7 ‚‚ Ð@Ñ s œ s@ ‰ 7 ‚ Now, for 0 − [ ‡ , we have Ð@ @Ð0 Ñ s ‰ 7 ‚ Ñ0 œ s@Ð7 ‚ 0 Ñ œ s@Ð0 ‰ 7 Ñ œ 0 Ð7 @Ñ œ 7ç and so 7 ‚‚ Ð@Ñ s œ 7ç@ Thus, 7 ‚‚ ‰ -ÐZ ÑÐ@Ñ œ -Ð[ ÑÐ7 @Ñ œ -Ð[ Ñ ‰ 7 Ð@Ñ and so 7 ‚‚ ‰ -ÐZ Ñ œ -Ð[ Ñ ‰ 7 If J À FinVect Ê FinVect is the functor that sends Z to Z ‡‡ and 7 À Z Ä [ to 7 ‚‚ À Z ‡‡ Ä [ ‡‡ , then this can be written Appendix A2: A Bit of Category Theory 291 J 7 ‰ -ÐZ Ñ œ -Ð[ Ñ ‰ M 7 where M is the identity functor on FinVect. But this says that the family of natural maps Ö-ÐZ Ñ× is a natural transformation from the identity functor M to the double-dual functor J . This accounts for the term natural map as it is applied to the map @ È s@. The only categorical result we will need is the following, which says essentially that functors act like ordinary functions with respect to surjectivity and injectivity. Theorem A2.8 1) Let J ß KÀ V Ê W be naturally isomorphic functors. a) J is faithful if and only if K is faithful. b) J is full if and only if K is full. † In particular, if J ¸ MV , then J is full and faithful and so the morphism maps J À homV ÐEß FÑ Ä homV ÐJ Eß J FÑ are bijections. 2) Let J À V Ê W and KÀ W Ê V be functors. a) If K ‰ J is faithful then J is faithful. b) If K ‰ J is full then K is full. In particular, if † K ‰ J ¸ MV † and J ‰ K ¸ MW then J and K are both full and faithful. † Proof. For part 1), let -À J ¸ K be a natural isomorphism. Assume that K is faithful. To see that J is faithful, if 0 ß 1 − homV ÐEß FÑ, then " " J 0 œ J 1 Ê -F ‰ J 0 ‰ -E œ -F ‰ J 1 ‰ -E Ê K0 œ K1 Ê0 œ1 To see that J is full, let 2 − homV ÐJ Eß J FÑ. Let " 5 œ -F ‰ 2 ‰ -E À KE Ä KF Since K is full, there is an 0 À E Ä F such that K0 œ 5 and so " " 2 œ -F ‰ 5 ‰ -E œ - F ‰ K0 ‰ -E œ J 0 Hence J is full. For part 2), if K ‰ J is faithful, then J is faithful, for if 0 ß 1 − homV ÐEß FÑ, we have 292 Lattices and Ordered Sets J 0 œ J 1 Ê KJ 0 œ KJ 1 Ê 0 œ 1 Also, if K ‰ J is full, then K is full, for if 2 − homV ÐEß FÑ, then there is an 0 À E Ä F such that 2 œ KJ 0 œ KÐJ 0 Ñ. Natural Isomorphism of Categories It seems reasonable to say that two categories V and W are isomorphic if there exist functors J À V Ê W and KÀ W Ê V for which J ‰ K œ MW and K ‰ J œ MV However, this notion turns out to be too restrictive. A more useful notion comes when we require only that the two compositions J ‰ K and K ‰ J are naturally isomorphic to the respective identity functors. Definition Two categories V and W are naturally isomorphic if there are functors J À V Ê W and KÀ W Ê V for which † J ‰ K ¸ MW † and K ‰ J ¸ MV where MW and MV are identity functors, as shown in Figure A2.5. When the functors J and K are contravariant, this condition is often expressed by saying that V and W are dual categories. A f λA GFA FGX B λB GFf GFB FGg κX X FGY κY g Y Figure A2.5 Example A2.9 As we will see in the text proper, the category FinDLat of finite distributive lattices is dual to the category FinPoset of finite posets. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] Balbes, R. and Dwinger, Ph., Distributive Lattices, University of Missouri Press, 1974. Banaschewski, B., On some theorems equivalent with the axiom of choice, Z. Math. Logik Grundlagen Math. 7, (1961), 279–282. Bell, J. L. and Fremlin, D. H., The maximal ideal theorem for lattices of sets, Bull. London Math. Soc. 4, (1972), 1–2. 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Index of Symbols Æ + œ ÖB − T ± B Ÿ +× Å + œ ÖB − T ± B +× ” : join • : meet “ : disjoint union ² : incomparable or parallel ö : covers « : covers or is equal to +Òαß" Ó : the set of relative complements of + with respect to Òαß " Ó E? : the set of all upper bounds for E Ej : the set of all lower bounds for E FÆ Ð+Ñ ³ F ∩ Æ + ConÐPÑ: the set of all congruence classes on P EpiÐPÑ: the set of all epimorphisms with domain P ÒEpiÐPÑÓ: the set of all equivalence classes of epimorphisms with domain P Y! : proper filters FLÐ\Ñ: the free lattice over \ +: the identity function \ ÐPÑ: ideals of P \! ÐPÑ: proper ideals of P \g ÐPÑ œ \ ÐPÑ ∪ Ög× ] ÐPÑ: the set of join-irreducible elements in P ]Æ Ð+Ñ ³ ] ÐPÑ ∩ Æ + ^ÐPÑ: the set of compact elements of P ^Æ Ð+Ñ ³ ^ÐPÑ ∩ Æ + ,: the kappa function defined by ,ÐBÑ œ ^Æ ÐBÑ j : the lower bound operator `ÐPÑ: the set of meet-irreducible elements in P Q$ : see Figure 3.4 R& : see Figure 3.4 NFÐ>Ñ: the normal form of > bÐT Ñ: the down-sets of T b‡ Ð[ Ñ ³ bÐ[ Ñ Ï Ö[ × 298 Lattices and Ordered Sets cc+ œ ÖT − SpecÐPÑ ± +  T × T Ò+Ó: initial segment of T at + kÐWÑ: power set of W k! : proper subsets StConÐPÑ: the set of all standard congruence relations on P g\ : the lattice terms over \ 9Æ : the down map defined by 9Æ + œ Æ + y B× 9cÅ : the not-up map defined by 9cÅ + œ ÖB − T ± + Ÿ 3: the rho function defined by 3Ð+Ñ œ cc+ hÐT Ñ: the up-sets of T ? : the upper bound operator ] ä \ : there is an injection from ] into \ ] Ç \ : there is a bijection from \ to ] Index absorption laws, 60 absorption, 60 alephs, 41 Alexander subbasis lemma, 277 algebra of sets, 131 algebra, 154 algebraic closure operator, 158 algebraic intersection-structures, 157 algebraic, 156 antichain cover, 25 antichain, 5 anti-embedding, 14 antisymmetric, 2 Arbitrary lattices, 149 Arguesian identity, 119 Arguesian, 119 arrows, 210, 281 ascending chain condition, 16 associativity, 51, 60 asymmetric, 2 atom, 7 atom, 80 atomic, 7, 80 atomistic, 80, 90 avoidance property, 170 axiom of choice, 8, 10 base, 172 base, 74, 276, 279 basis, 241, 276 binary relation, 1 binary, 153 blocks, 110 Boolean algebra, 130, 149 Boolean homomorphism, 134 Boolean lattice, 129 Boolean prime ideal theorem, 174 Boolean ring, 131 Boolean space, 218 Boolean subalgebra, 131 bounded, 7 Brouwerian lattice, 127 canonical forms, 255 canonical projection, 191 Cantor's theorem, 40 Cantor–Schröder–Bernstein theorem, 40 cardinal arithmetic, 43 cardinal, 38 cardinality, 38 carrier set, 79 cartesian product, 1 category, 210, 281 chain cover, 18 chain, 5 chain-complete, 43, 68 choice function, 8 clopen, 275 closed cylinder, 279 closed initial segment, 10 closed interval, 10 closed set, 275 closed, 55 closure operator, 55 closure systems, 55 closure, 55 300 Lattices and Ordered Sets CLP, 163 coalesce, 66, 67 coalescing pair, 66, 67 coarser, 111 coatom, 7 cofinal, 46 cofinite, 54, 131 coinitial, 46 cokernel, 204 commutativity, 51, 60 commute, 187 compact, 155, 276 compactly-generated, 156 comparable, 3 compatible, 21 complement, 82 complementary, 50, 82 complemented lattice, 82 complete atomic Boolean algebras, 150 complete lattice, 53 complete sublattice, 68 complete, 43, 54, 262 completely distributive, 140 completely meet-irreducible, 163 completion by cuts, 86 completion, 85 component, 214, 286 composition, 281 congruence classes, 180 congruence kernel, 191, 192 congruence lattice problem, 163 congruence lattice, 162, 185 congruence relation, 162, 179, 180, 243 conjunctive, 142 connected, 276 consistent, 273 consistently complete, 273 contained in, 3 contains, 3 continuous, 158, 261, 278 continuum hypothesis, 41 contravariant functor, 212, 283 convex, 10 copoint, 7 correspondence theorem, 194 covariant functor, 212, 283 covers, 4 CPO, 43, 54, 262 cylinder, 172, 279 de Morgan's laws, 129 Dedekind–MacNeille (DM) completion, 86 Dedekind's law, 97 degenerate, 95 descending chain condition, 16 diamond, 98 Dilworth's theorem, 18 direct product, 54 directed join, 155 directed, 6 discrete topology, 275 disjoint, 18 disjunctive, 91, 142 distance function, 113 distance, 15 distributive lattice, 81, 149 distributive laws, 80, 96 distributive prime ideal theorem, 174 distributive, 96, 205 double lower covering condition, 109 double upper covering condition, 109 down closure, 14 down map, 14, 15 down-set functor, 213, 215, 285 down-set, 13 dual categories, 214, 290 dual poset, 10 dual statement, 11 duality principle for posets, 12 embedded, 13 equational class, 61, 240 equinumerous, 37 equipollent, 37 equipotent, 37 equivalence class, 2 Index equivalence relation, 2 equivalent up to associativity and commutativity, 252 evaluation map, 70, 240 existence property, 170 extends, 114 extension property, 170 extension theorem, 174 extension, 116, 117 extensive, 55, 58 faithful, 79, 213, 284 field of sets, 131 filter generated, 76 filter, 75 finer, 111 finitary operation, 153 finitary, 57 finite character, 9 finite, 33 finite-cofinite algebra, 54, 131 finite-join-dense, 70 finitely-generated, 154, 157 finite-meet-dense, 70 first isomorphism theorem, 194, 203 fixed point, 261, 262 forgetful functors, 285 formal lattice terms, 244 free lattice, 241 freely generated, 241 friendly, 195 Frink's condition, 143 full, 213, 284 functor on, 212, 284 functor, 211, 283 Galois connection, 58 Galois correspondence, 58 Galois group, 58 generalized continuum hypothesis, 42 graded, 16 grading function, 16 greater than or equal to, 3 greatest lower bound, 7 301 greatest, 6 Hall's condition, 19 Hartogs number, 40 has arbitrary meets, 49 has arbitrary nonempty meets, 49 has finite meets, 49 has, 10 Hasse diagram, 4 Hausdorff maximality principle, 9, 10 Hausdorff, 276 height, 15 Heyting algebra, 127 Heyting lattice, 127 hom set, 211 homeomorphism, 72, 278 homomorphism theorem, 194 ideal generated, 76 ideal kernel, 78, 190, 192 ideal, 74 idempotency, 51, 60 idempotent, 55, 131 identity morphism, 211, 281 image, 36 incomparable, 3 increasing witness sequence, 185 increasing, 270 index ordinal, 36 indicator function, 78 induced inverse map, 135 induced topology, 275 inductive hypothesis, 37 inductive, 43 infimum, 7 infinitary, 57, 153 infinite, 33 infinitely distributive, 139 inflationary, 28, 261 inherits, 49 initial ordinal, 38 initial segment, 10, 27 intersection-structure, 55 invariant, 263 inverse, 281 302 Lattices and Ordered Sets irredundant joinirreducible representation, 123 irredundant, 123, 164 irreflexive, 2 isomorphic, 281 isomorphism, 211, 281 isotone, 13 join maps, 103 join, 7 join-continuous, 158 join-cover, 155 join-dense, 70 join-homomorphism, 73 join-inaccessible, 166 join-infinitely distributive, 139 join-irreducible, 64 join-normal form, 251 join-prime, 50 join-semilattice, 51 Jordan–Dedekind chain condition, 15 kappa function, 225 kernel equivalent, 203, 231 kernel, 190 Knaster–Tarski fixed point theorem, 264 Kurosh–Ore replacement property, 123 Kurosh–Ore theorem, 123 largest, 6 lattice embedding, 72 lattice endomorphism, 72 lattice epimorphism, 72 lattice equation, 240 lattice generated, 126 lattice homomorphism, 72 lattice identity, 240 lattice isomorphism, 72 lattice monomorphism, 72 lattice polynomial, 69, 239 lattice term, 69, 239 lattice, 53 least upper bound, 7 least, 6 length, 5, 111 less than or equal to, 3 lexicographic order, 4 limit ordinal, 35 linear kernel, 25 linear sum, 23 linear, 3 logically equivalent, 11 lower bound, 7 lower covering condition, 106 lower MacNeille extension, 92 lower semimodular, 106 MacNeille completion, 86 MacNeille extensions, 92 maps, 210, 281 matrix representation functor, 287 maximal antichain, 6 maximal chain, 5 maximal condition, 17 maximal extension theorem, 173 maximal ideal theorem, 173 maximal separation property, 169 maximal separation theorem, 173 maximal, 6, 77, 170 maximum, 6 median property, 125 mediating morphism, 193, 241 meet maps, 103 meet representation, 113 meet, 7 meet-continuous, 166 meet-dense, 70 meet-homomorphism, 73 meet-infinitely distributive, 139 meet-irreducible, 64 meet-normal form, 251 meet-prime, 50 meet-representation, 164 meet-semilattice, 51 meet-structure, 55 minimal condition, 17 minimal element, 6 minimum, 6 modular law, 81, 96 Index modular, 81, 96 monotone, 13, 55 morphism function, 283 morphisms, 210, 281 natural isomorphism, 214, 286 natural projection, 191 natural transformation, 213, 286 naturally equivalent, 214 naturally isomorphic, 214, 290 nontrivial lattice, 54 nontrivial normal form, 251 normal completion, 86 normal form, 251 nullary, 153 object function, 283 objects, 210, 281 open cover, 276 open cylinder, 279 open map, 279 open sets, 275 operation, 153 order embedded, 13 order embedding, 13 order filter, 13 order ideal, 13 order isomorphism, 13 order type, 35 order, 2 ordered Boolean space, 218 ordered topological space, 218 order-homeomorphism, 218 order-preserving, 13 order-reversing, 58 ordinal arithmetic, 42 ordinal number, 31 ordinal, 31 parallel, 3 partial functions, 3 partial order, 2 partially ordered set, 2 partition, 110 pentagon, 98 Philip Hall's theorem, 19 303 point, 7 poset, 2 possesses, 10 post-fixed point, 262 power set algebra, 131 power set functor, 212, 285 power set subalgebra, 131 power set sublattice, 69 pre-fixed point, 262 preorder, 3 Priestley dual space, 220 Priestley duality, 222, 227 Priestley prime ideal space, 220 Priestley representation theorem for bounded distributive lattices, 226 Priestley topological space, 218 prime point-separation property, 175 prime-separation property, 169 prime separation theorem, 175 prime, 77, 170, 175 principal congruence generated by, 204 principal filter generated by, 14 product order, 3 product topology, 278 product, 3, 187 proper cylinders, 279 proper filter, 75 proper ideal, 74 proper join, 252 property of posets, 10 pseudocomplement, 126, 231 pseudocomplemented lattice, 126 quadrilateral property, 184 quadrilateral, 95 quasiorder, 3 quotient lattice, 189 range, 36 reachable, 111 refinement, 111 refinement, 3 refines, 250 304 Lattices and Ordered Sets reflexive, 2 relative complement, 83 relative pseudocomplement, 126 relatively complemented, 84 relatively free, 241 relatively pseudocomplemented, 127 representation map, 79 representation theorem for atomic Boolean algebras, 146 representation theorem for Boolean algebras, 149, 209 representation theorem for distributive lattices with DCC, 146 representation theorem for distributive lattices, 149, 209 representation theorem for lattices, 79 representation theorems, 149 representation, 79, 112 representative, 250 rho map, 148 ring of sets, 69 satisfies, 10 SDR, 1 sectionally complemented, 84 self-dual, 11 semicomplement, 126 semidistributive laws, 260 semilattice, 51 separation property, 175 separation theorems, 169 separation, 276 sequence, 36, 111, 185 simple, 202 smallest, 6 smooth, 92 SMP, 136 SMP1, 136 SMP2, 136 spectrum functor, 212, 215, 285 spectrum, 77 standard, 195, 197 statement about posets, 10 Stone dual space, 220, 236 Stone duality for Boolean algebras, 230 Stone prime ideal space, 220, 236 Stone representation theorem for Boolean algebras, 230 Stone space, 218 strict order, 4, 23 strict partial order, 4 strictly increasing, 270 strictly meet-irreducible, 163 strictly monotone, 13 strongly atomic, 80 subalgebra generated by, 154 subalgebra lattice, 154 subalgebra, 154 subbase, 276 subbasis, 276 subcategory, 211 subcategory, 283 subcover, 276 sub-CPO, 262 subdirect representation, 194 sublattice generated, 69 sublattice, 68 subsequence, 36 subspace, 275 successor ordinal, 35 successor, 34 supremum, 7 symmetric closure, 20 symmetric, 2 system of distinct representatives, 2 term, 69, 239 topological sorting, 21 topological space, 275 topology, 275 total, 3 totally intransitive, 20 totally ordered, 3 totally order-separated, 218 totally separated, 218, 276 Transfinite induction: version 1, 37 Transfinite induction: Index version 2, 37 transfinite sequence, 36 transitive closure, 20 transitive, 2, 31 trivial lattice, 54 trivial normal form, 251 trivial variety, 240 trivial, 110 Tukey's lemma, 9, 10 two-sided, 281 type, 112, 113 ultrafilter, 77 unary, 153 underlying set, 36 uniquely complemented lattice, 82 uniquely relatively complemented, 84 unit element, 6 universal algebra, 154 universal property of quotients, 193 universal, 193, 241 unsolvable, 247 up closure, 14 up map, 14, 15 upper bound, 7 upper covering condition, 105 upper MacNeille extension, 92 upper semimodular, 106 up-set, 13 variable, 69, 251, 239, 241 variety, 61, 240 weakly atomic, 80 weakly preserves covering, 105 weight, 69, 239, 251 well-orderable, 9, 27 well-ordered set, 9, 27 well-ordering principle, 9, 10 well-ordering, 9, 27 width, 5 witness, 111, 185 word problem, 247 zero element, 7 Zorn's lemma, 9, 10 305
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